Since their origin - 250 years ago - partial differential equations form, besides modern stochastics, the most adequate and powerful instrument to provide a mathematical model for nature. This is reflected in the treatment of real world problems at the Weierstrass Institute. Unfortunately, in realistic situations the data of the problems, such as the domain under consideration, as well as the coefficients are mostly nonsmooth. Moreover, the occurring boundary conditions are often of mixed type. This prevents the straight forward application of classical methods in partial differential equations. Beginning around 1990, however, there was a considerable advance in finding ideas how to treat also equations which include nonsmooth data. At the Weierstrass Institute this research has two essential focuses:
  • Regularity for the solutions of linear elliptic equations
  • Existence, uniqueness and regularity for evolution equations
Examples of non-smooth domains: A non-Lipschitz domain, Dirichlet boundaries in gray (left), The double beam is the the prototype of a domain which is Lipschitzian, but not strong Lipschitzian (right)

Regularity for the solutions of linear elliptic equations

The emphasis in this topic is on the integrability of the gradients of solutions up to an index which is larger than the dimension of the underlying space and, on the other hand, Hölder continuity of the solution under very weak assumptions on the data of the problem. The motivation concerning the integrability property is the insight that exactly this would deliver uniqueness for the solution for many nonlinear models and, additionally, ideas for the approximation of it. The starting point was the result of Gröger, who proved that this can always be achieved in a very general setting if the space dimension equals two. Since then several results have been proved in 3D which partly were already used in several application areas of the institute. The final goal is to establish an appropriate global setting which includes specific discontinuities of the above mentioned type and, nevertheless, assures the required integrability in three dimensions. (It is clear that, firstly, the three dimensional case is the essential one for applications in real space, and, secondly, that the required integrability property cannot be expected in higher dimensions e.g. in case of mixed boundary conditions.) The impetus for the investigation of Hölder continuity came from optimal control, where the corresponding results serve by now as a backbone in the treatment of control problems with pointwise inequality constraints on the state.
A typical non-smooth domain: Steel hardening in rack-and-pinion actuators, to be found e.g. in steering mechanisms, see Meinlschmidt-Meyer-Rehberg 2016.

Existence, uniqueness and regularity for evolution equations

The interest here comes from problems out of semiconductor modeling, elasticity/plasticity/damage modeling, and battery modeling (ion transfer through electrolytes + electrolyte/electrode interaction) at WIAS. Concerning reaction-diffusion equations, an idea of Clement and Li, generalized by Prüss, shows (local in time) existence and uniqueness for these equations, provided one knows maximal parabolic regularity for the - linear - elliptic operator which corresponds to the initial time point and the initial value. Thus, maximal parabolic regularity was proved for general second order divergence operators on different Banach spaces: first on all Lp spaces, thus obtaining an instrument for the treatment of equations where the Neumann boundary condition is homogeneous and no distributional right hand sides occur. Second, it was shown -together with the Weierstrass awardee Dr. Haller-Dintelmann -that maximal parabolic regularity holds true for elliptic divergence operators in a huge variety of distributional spaces which allow an adequate treatment of the corresponding nonlinear parabolic equations, among them nonlinear heat equations and Van Roosbroeck's system in 3D.

Highlights

Global-in-time existence analysis for improved Nernst-Planck-Poisson models

These PDE-models consist of convection-diffusion-reaction equations describing the mass-transfer of chemical species, the equations of momentum balance for the barycentric velocity and the pressure of the mixture (compressible/incompressible Navier-Stokes equations) and the Poisson equation for the electrical potential. The PDEs in the bulk are supplied with complex boundary conditions describing the surface adsorption of species and their reaction with the components of adjacent materials. The constitutive equations for the diffusion fluxes and the reaction rates are derived by means of modern thermodynamic methods that guarantee consistency with the entropy principle and lead to new PDE structures. Several challenges occur in the analysis of such systems:
  • The Onsager matrix possesses a singular eigenvalue
  • Cross-diffusion
  • Highly nonlinear reaction rates
  • Pressure contribution to the chemical potentials, pressure coupling of the diffusion fluxes
  • Compressible Navier-Stokes system
Exploiting the natural energy/dissipation estimates, fundamental methods of convex analysis and the Lions compactness method for the Navier-Stokes operator, convergent approximation schemes were constructed and the existence of global-in-time weak solutions for this very complex system was proved. This represents an absolute breakthrough in the analysis of similar models. Also qualitative questions like the vanishing of species or further degeneracy of the Onsager matrix were discussed.
Solutions of p(x)-Laplace equation in 2D with constant right-hand side and homogeneous Dirichlet boundary conditions for. The value of p in the back is fixed to 2 while it is varying in the front.

Existence and regularity of solutions to thermistor systems for organic devices

Organic semiconductor devices are carbon-based electronic devices that show a strong interplay between current and heat flow due to temperature-activated hopping transport on one hand and Joule self-heating on the other hand. On an empirical level, this electrothermal feedback can be described by means of a p(x)-Laplace thermistor model for the electrostatic potential and the temperature, where the p(x)-Laplacian takes the non-Ohmic behavior of the individual organic layers into account and the temperature dependence is given by an Arrhenius-like law. A mathematical challenge arises from the fact that the Joule self-heating term in the heat equation is a priori only in L1.

Due to missing log-Hölder continuity of the exponent p(x), known techniques for the treatment of the p(x)-Laplacian fail in the setting of organic devices. However, a novel higher integrability result for the gradient of the electrostatic potential was proved in Glitzky-Liero 2017 by means of localization, Caccioppoli estimates, and Gehring-type arguments. The resulting higher integrability of the Joule heat term in the heat flow equation then allows to prove the solvability of the p(x)-Laplace thermistor problem by Schauder's fixed-point theorem. A second approach is presented in Bulíček-Glitzky-Liero 2016a. Therein, higher spatial dimensions and arbitrary exponents p(x) are considered. The existence proof is based on a Galerkin approximation of a regularized version of the thermistor system, where the crucial Joule heat term is approximated such that it remains bounded. Finally, in Bulíček-Glitzky-Liero 2016b the notion of entropy solutions of the heat equation in conjunction with Schauder's fixed point theorem was investigated. Uniqueness of solutions for such systems cannot be expected due to the hysteretic behavior.

Publications

  Monographs

  • C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, Springer International Publishing Switzerland, Cham, 2016, xii+155 pages, (Monograph Published).
    Abstract
    Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance.

  • A. Mielke, Chapter 3: On Evolutionary $Gamma$-Convergence for Gradient Systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, A. Muntean, J.D.M. Rademacher, A. Zagaris, eds., 3 of Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing Switzerland, Cham, 2016, pp. 187--249, (Chapter Published).
    Abstract
    In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional Eε and the dissipation potential Rε or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions uε. We investigate the question under which conditions the limits u of (subsequences of) the solutions uε are solutions of the gradient system generated by the Γ-limits E0 and R0. Here the choice of the right topology will be crucial as well as additional structural conditions.
    We cover classical gradient systems, where Rε is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.

  • A. Mielke, T. Roubíček, Rate-independent Systems. Theory and Application, 193 of Applied Mathematical Sciences, Springer International Publishing, New York, 2015, vii+660 pages, (Monograph Published).
    Abstract
    This monograph provides both an introduction to and a thorough exposition of the theory of rate-independent systems, which the authors have been working on with a lot of collaborators over 15 years. The focus is mostly on fully rate-independent systems, first on an abstract level either with or even without a linear structure, discussing various concepts of solutions with full mathematical rigor. Then, usefulness of the abstract concepts is demonstrated on the level of various applications primarily in continuum mechanics of solids, including suitable approximation strategies with guaranteed numerical stability and convergence. Particular applications concern inelastic processes such as plasticity, damage, phase transformations, or adhesive-type contacts both at small strains and at finite strains. A few other physical systems, e.g. magnetic or ferroelectric materials, and couplings to rate-dependent thermodynamic models are considered as well. Selected applications are accompanied by numerical simulations illustrating both the models and the efficiency of computational algorithms. In this book, the mathematical framework for a rigorous mathematical treatment of "rate-independent systems" is presented in a comprehensive form for the first time. Researchers and graduate students in applied mathematics, engineering, and computational physics will find this timely and well written book useful.

  • A. Mielke, Chapter 5: Variational Approaches and Methods for Dissipative Material Models with Multiple Scales, in: Analysis and Computation of Microstructure in Finite Plasticity, S. Conti, K. Hackl, eds., 78 of Lecture Notes in Applied and Computational Mechanics, Springer International Publishing, Heidelberg et al., 2015, pp. 125--155, (Chapter Published).
    Abstract
    In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit.
     
    As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.

  • E. Valdinoci, ed., Contemporary PDEs between theory and applications, 35 of Discrete and Continuous Dynamical Systems Series A, American Institute of Mathematical Sciences, Springfield, 2015, 625 pages, (Collection Published).

  • B. Fiedler, M. Haragus, A. Mielke, G. Raugel, Y. Yi, eds., Special Issue in Memoriam of Klaus Kirchgässner, 27 of J. Dynam. Differential Equations, Springer International Publishing, Cham et al., 2015, 674 pages, (Collection Published).

  • A. Mielke, Chapter 21: Dissipative Quantum Mechanics Using GENERIC, in: Recent Trends in Dynamical Systems -- Proceedings of a Conference in Honor of Jürgen Scheurle, A. Johann, H.-P. Kruse, F. Rupp, S. Schmitz, eds., 35 of Springer Proceedings in Mathematics & Statistics, Springer, Basel et al., 2013, pp. 555--585, (Chapter Published).
    Abstract
    Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework (General Equations for Non-Equilibrium Reversible Irreversible Coupling) to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition. One of our models couples a quantum system to a finite number of heat baths each of which is described by a time-dependent temperature. The dissipation mechanism is modeled via the canonical correlation operator, which is the inverse of the Kubo-Mori metric for density matrices and which is strongly linked to the von Neumann entropy for quantum systems. Thus, one recovers the dissipative double-bracket operators of the Lindblad equations but encounters a correction term for the consistent coupling to the dissipative dynamics. For the finite-dimensional and isothermal case we provide a general existence result and discuss sufficient conditions that guarantee that all solutions converge to the unique thermal equilibrium state. Finally, we compare of our gradient flow formulation for quantum systems with the Wasserstein gradient flow formulation for the Fokker-Planck equation and the entropy gradient flow formulation for reversible Markov chains.

  • G. Dal Maso, A. Mielke, U. Stefanelli, eds., Rate-independent Evolutions, 6 (No. 1) of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Sciences, Springfield, 2013, 275 pages, (Collection Published).

  • A. Mielke, F. Otto, G. Savaré, U. Stefanelli, eds., Variational Methods for Evolution, 8 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2011, pp. 3145--3216, (Chapter Published).

  • P. Colli, A. Damlamian, N. Kenmochi, M. Mimura, J. Sprekels, eds., Proceedings of International Conference on: Nonlinear Phenomena with Energy Dissipation: Mathematical Analysis, Modeling and Simulation, 29 of Gakuto International Series Mathematical Sciences and Applications, Gakkōtosho, Tokyo, 2008, 475 pages, (Collection Published).

  • U. Bandelow, H. Gajewski, R. Hünlich, Chapter 3: Fabry--Perot Lasers: Thermodynamics-based Modeling, in: Optoelectronic Devices --- Advanced Simulation and Analysis, J. Piprek, ed., Springer, New York, 2005, pp. 63-85, (Chapter Published).

  • A. Mielke, Chapter 6: Evolution of Rate-independent Systems, in: Handbook of Differential Equations. Vol. 2: Evolutionary Equations, C. Dafermos, E. Feireisl, eds., Elsevier Science B.V., Amsterdam, 2005, pp. 461-559, (Chapter Published).

  • B. Fiedler, K. Gröger, J. Sprekels, eds., EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, 1, World Scientific, Singapore [u. a.], 2000, 806 pages, (Monograph Published).

  • B. Fiedler, K. Gröger, J. Sprekels, eds., EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, 2, World Scientific, Singapore [u. a.], 2000, 681 pages, (Monograph Published).

  Articles in Refereed Journals

  • K. Disser, J. Rehberg, A.F.M. TER Elst, Hölder estimates for parabolic operators on domains with rough boundary, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XVII (2017) pp. 65--79.
    Abstract
    In this paper we investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove Hölder continuity of the solution in space and time.

  • M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017) pp. 1--35.
    Abstract
    Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a Cahn--Hilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017) pp. 37--54.
    Abstract
    This paper is concerned with a phase field system of Cahn--Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates

  • J. Sprekels, E. Valdinoci, A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation, SIAM Journal on Control and Optimization, 55 (2017) pp. 70--93.
    Abstract
    In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the power of a positive definite operator having a positive and discrete spectrum. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the ``control parameter'' that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new classof identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter also the domain of definition, and thus the underlying function space, of the fractional operator changes.

  • A. Glitzky, M. Liero, Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 34 (2017) pp. 536--562.
    Abstract
    We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a $p(x)$-Laplace-type problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the $p(x)$-Laplacian. Finally, Schauder's fixed-point theorem is used to show the existence of solutions.

  • N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, A review of variational multiscale methods for the simulation of turbulent incompressible flows, Archives of Computational Methods in Engineering. State of the Art Reviews, 24 (2017) pp. 115--164.
    Abstract
    Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier--Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

  • E. Cinti, J. Davila, M. Del Pino, Solutions of the fractional Allen--Cahn equation which are invariant under screw motion, Journal of the London Mathematical Society. Second Series, 94 (2016) pp. 295--313.
    Abstract
    We establish existence and non-existence results for entire solutions to the fractional Allen--Cahn equation in R3 , which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.

  • K. Disser, G.P. Galdi, G. Mazzone, P. Zunino, Inertial motions of a rigid body with a cavity filled with a viscous liquid, Archive for Rational Mechanics and Analysis, 221 (2016) pp. 487--526.
    Abstract
    We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. Zhukovskiy's Theorem states that in the limit of time going to infinity, the relative fluid velocity tends to 0 and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. We give a rigorous proof of this result.
    In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.

  • A. Alphonse, Ch.M. Elliott, Well-posedness of a fractional porous medium equation on an evolving surface, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 137 (2016) pp. 3--42.

  • M. Cozzi, A. Farina, E. Valdinoci, Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Advances in Mathematics, 293 (2016) pp. 343--381.
    Abstract
    We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classi?cation results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non- degenerate anisotropic equations.

  • S.P. Frigeri, Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016) pp. 1957--1993.
    Abstract
    We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists of a Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

  • M. Heida, B. Schweizer, Non-periodic homogenization of infinitesimal strain plasticity equations, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016) pp. 5--23.

  • M. Liero, A. Mielke, G. Savaré, Optimal transport in competition with reaction: The Hellinger--Kantorovich distance and geodesic curves, SIAM Journal on Mathematical Analysis, 48 (2016) pp. 2869--2911.
    Abstract
    We discuss a new notion of distance on the space of finite and nonnegative measures on Ω ⊂ ℝ d, which we call Hellinger-Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich-Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Ω. We give a construction of geodesic curves and discuss examples and their general properties.

  • D. Peschka, N. Rotundo, M. Thomas, Towards doping optimization of semiconductor lasers, Journal of Computational and Theoretical Transport, 45 (2016) pp. 410--423.
    Abstract
    We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum-well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.

  • M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M.E. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptotic Analysis, 97 (2016) pp. 189--210.
    Abstract
    We study a PDE system describing the motion of liquid crystals by means of the Q?tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate (1 + t)?? as t ? ? 1 for a certain ? > 2 .

  • S. Dipierro, O. Savin, E. Valdinoci, Graph properties for nonlocal minimal surfaces, Calculus of Variations and Partial Differential Equations, 55 (2016) pp. 86/1--86/25.
    Abstract
    In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler?Lagrange equation related to the nonlocal mean curvature.

  • S. Patrizi, E. Valdinoci, Relaxation times for atom dislocations in crystals, Calculus of Variations and Partial Differential Equations, 55 (2016) pp. 71/1--71/44.
    Abstract
    We study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls?Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. In this sense, the system exhibits two different decay behaviors, namely an exponential time decay versus a polynomial decay in the space variables (and these two homogeneities are kept separate during the time evolution).

  • P. Bringmann, C. Carstensen, Ch. Merdon, Guaranteed error control for the pseudostress approximation of the Stokes equations, Numerical Methods for Partial Differential Equations. An International Journal, 32 (2016) pp. 1411--1432.
    Abstract
    The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in $L^2$. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.

  • M. Bulíček, A. Glitzky, M. Liero, Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents, SIAM Journal on Mathematical Analysis, 48 (2016) pp. 3496--3514.
    Abstract
    We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L1 term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the L1 right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.

  • P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 73 (2016) pp. 195--225.
    Abstract
    A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.

  • P. Colli, G. Gilardi, J. Sprekels, Constrained evolution for a quasilinear parabolic equation, Journal of Optimization Theory and Applications, 170 (2016) pp. 713--734.
    Abstract
    In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the Cauchy--Neumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set K of L2(Ω). Then, we consider convex sets of obstacle or double-obstacle type, and we can act on the factor of the feedback control in order to be able to reach the convex set within a finite time, by proving rigorously this property.

  • P. Colli, G. Gilardi, J. Sprekels, Distributed optimal control of a nonstandard nonlocal phase field system, AIMS Mathematics, 1 (2016) pp. 246--281.
    Abstract
    We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

  • P. Colli, G. Gilardi, J. Sprekels, On an application of Tikhonov's fixed point theorem to a nonlocal Cahn--Hilliard type system modeling phase separation, Journal of Differential Equations, 260 (2016) pp. 7940--7964.
    Abstract
    This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006) 105-118. The model consists of an initial-boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter ρ and the chemical potential μ. Singular contributions to the local free energy in the form of logarithmic or double-obstacle potentials are admitted. In contrast to the local model, which was studied by P. Podio-Guidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long-range interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space.

  • J. Dávila, M. Del Pino, S. Dipierro, E. Valdinoci, Nonlocal Delaunay surfaces, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 137 (2016) pp. 357--380.
    Abstract
    We construct codimension 11 surfaces of any dimension that minimize a periodic nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

  • A. Farina, E. Valdinoci, 1D symmetry for semilinear PDEs from the limit interface of the solution, Communications in Partial Differential Equations, 41 (2016) pp. 665--682.
    Abstract
    We study bounded, monotone solutions of ?u = W?(u) in the whole of ?n, where W is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, u is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of ?-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and wishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.

  • R. Haller-Dintelmann, A. Jonsson, D. Knees, J. Rehberg, Elliptic and parabolic regularity for second order divergence operators with mixed boundary conditions, Mathematical Methods in the Applied Sciences, 39 (2016) pp. 5007--5026.
    Abstract
    We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces.

  • H. Meinlschmidt, J. Rehberg, Hölder-estimates for non-autonomous parabolic problems with rough data, Evolution Equations and Control Theory, 5 (2016) pp. 147--184.
    Abstract
    In this paper we establish Hölder estimates for solutions to non-autonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations.

  • X. Ros-Oton, E. Valdinoci, The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains, Advances in Mathematics, 288 (2016) pp. 732--790.
    Abstract
    We study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample.

  • S. Reichelt, Error estimates for elliptic equations with not exactly periodic coefficients, Advances in Mathematical Sciences and Applications, 25 (2016) pp. 117--131.
    Abstract
    This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.

  • N. Ahmed, G. Matthies, Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent convection-diffusion-reaction equations, Journal of Scientific Computing, 67 (2016) pp. 988--1018.
    Abstract
    This paper considers the numerical solution of time-dependent convection-diffusion-reaction equations. We shall employ combinations of streamline-upwind Petrov-Galerkin (SUPG) and local projection stabilization (LPS) methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin (dG) methods and continuous Galerkin-Petrov (cGP) methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. Finally the long-time behavior of overshoots and undershoots is investigated.

  • M.H. Farshbaf Shaker, C. Hecht, Optimal control of elastic vector-valued Allen--Cahn variational inequalities, SIAM Journal on Control and Optimization, 54 (2016) pp. 129--152.
    Abstract
    In this paper we consider a elastic vector-valued Allen--Cahn MPCC (Mathematical Programs with Complementarity Constraints) problem. We use a regularization approach to get the optimality system for the subproblems. By passing to the limit in the optimality conditions for the regularized subproblems, we derive certain generalized first-order necessary optimality conditions for the original problem.

  • S.P. Frigeri, E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal Cahn--Hilliard/Navier--Stokes system in two dimensions, SIAM Journal on Control and Optimization, 54 (2016) pp. 221 -- 250.
    Abstract
    We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier-Stokes system with a convective nonlocal Cahn-Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.

  • A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of Non-Equilibrium Thermodynamics, 41 (2016) pp. 141--149.
    Abstract
    Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

  • A. Mielke, T. Roubíček, Rate-independent elastoplasticity at finite strains and its numerical approximation, Mathematical Models & Methods in Applied Sciences, 26 (2016) pp. 2203--2236.
    Abstract
    Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions.
     
    In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

  • A. Mielke, R. Rossi, G. Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, Journal of the European Mathematical Society (JEMS), 18 (2016) pp. 2107--2165.
    Abstract
    Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent ows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for in nite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their di erentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on re ned lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.

  • R.I.A. Patterson, Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries, Journal of Evolution Equations, 16 (2016) pp. 261--291.
    Abstract
    Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.

  • K. Disser, M. Meyries, J. Rehberg, A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces, Journal of Mathematical Analysis and Applications, 430 (2015) pp. 1102--1123.
    Abstract
    In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.

  • K. Disser, M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks and Heterogeneous Media, 10 (2015) pp. 233-253.
    Abstract
    We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then, we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.

  • K. Disser, H.-Chr. Kaiser, J. Rehberg, Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems, SIAM Journal on Mathematical Analysis, 47 (2015) pp. 1719--1746.
    Abstract
    On bounded three-dimensional domains, we consider divergence-type operators including mixed homogeneous Dirichlet and Neumann boundary conditions and discontinuous coefficient functions. We develop a geometric framework in which it is possible to prove that the operator provides an isomorphism of suitable function spaces. In particular, in these spaces, the gradient of solutions turns out to be integrable with exponent larger than the space dimension three. Relevant examples from real-world applications are provided in great detail.

  • K. Disser, Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions, Analysis. International Mathematical Journal of Analysis and its Applications, 35 (2015) pp. 309--317.

  • S. Patrizi, E. Valdinoci, Crystal dislocations with different orientations and collisions, Archive for Rational Mechanics and Analysis, 217 (2015) pp. 231--261.
    Abstract
    We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time.

  • S. Patrizi, E. Valdinoci, Homogenization and Orowan's law for anisotropic fractional operators of any order, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 119 (2015) pp. 3--36.
    Abstract
    We consider an anisotropic fractional operator and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the fractional parameter. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowan's law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.

  • E. Rocca, R. Rossi, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM Journal on Mathematical Analysis, 74 (2015) pp. 2519--2586.
    Abstract
    In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and dam- age phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L^1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as ``entropic'', where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its ``entropic'' formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.

  • E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective Cahn--Hilliard equation by the velocity in three dimensions, SIAM Journal on Control and Optimization, 53 (2015) pp. 1654--1680.
    Abstract
    In this paper we study a distributed optimal control problem for a nonlocal convective Cahn-Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem.

  • P.-É. Druet, Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations, Discrete and Continuous Dynamical Systems, 8 (2015) pp. 479--496.
    Abstract
    We show that Lp vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We investigate the regularity of the solution fields for the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation 'in H' of the problem, in a reference geometrical setting allowing for material heterogeneities.

  • P.-É. Druet, Some mathematical problems related to the second order optimal shape of a crystallization interface, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 2443--2463.
    Abstract
    We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient.

  • S.P. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015) pp. 1257--1293.
    Abstract
    We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.

  • M. Heida, Existence of solutions for two types of generalized versions of the Cahn--Hilliard equation, Applications of Mathematics, 60 (2015) pp. 51--90.
    Abstract
    We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration u, gradient of concentration ?u and the chemical potential ?u?s?(u). The existence is shown using a newly developed generalization of gradient flows by the author and the theory of Young measures.

  • M. Heida, On systems of Cahn--Hilliard and Allen--Cahn equations considered as gradient flows in Hilbert spaces, Journal of Mathematical Analysis and Applications, 423 (2015) pp. 410--455.

  • M. Liero, Th. Koprucki, A. Fischer, R. Scholz, A. Glitzky, p-Laplace thermistor modeling of electrothermal feedback in organic semiconductors, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015) pp. 2957--2977.
    Abstract
    In large-area Organic Light-Emitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered. We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem.

  • S. Yanchuk, L. Lücken, M. Wolfrum, A. Mielke, Spectrum and amplitude equations for scalar delay-differential equations with large delay, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 537--553.
    Abstract
    The subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.

  • S. Yanchuk, G. Giacomelli, Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 92 (2015) pp. 042903/1--042903/12.
    Abstract
    Dynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms.

  • E. Bonetti, Ch. Heinemann, Ch. Kraus, A. Segatti, Modeling and analysis of a phase field system for damage and phase separation processes in solids, Journal of Partial Differential Equations, 258 (2015) pp. 3928--3959.
    Abstract
    In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced system, we start with a regularized version. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.

  • A. Di Castro, M. Novaga, R. Berardo, E. Valdinoci, Nonlocal quantitative isoperimetric inequalities, Calculus of Variations and Partial Differential Equations, 54 (2015) pp. 2421--2464.

  • S. Dipierro, E. Valdinoci, A density property for fractional weighted Sobolev spaces, Rendiconti Lincei -- Matematica e Applicazioni, 26 (2015) pp. 397--422.
    Abstract
    In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

  • S. Dipierro, E. Valdinoci, On a fractional harmonic replacement, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 3377--3392.
    Abstract
    Given $s ∈(0,1)$, we consider the problem of minimizing the Gagliardo seminorm in $H^s$ with prescribed condition outside the ball and under the further constraint of attaining zero value in a given set $K$. We investigate how the energy changes in dependence of such set. In particular, under mild regularity conditions, we show that adding a set $A$ to $K$ increases the energy of at most the measure of $A$ (this may be seen as a perturbation result for small sets $A$). Also, we point out a monotonicity feature of the energy with respect to the prescribed sets and the boundary conditions.

  • S. Dipierro, O. Savin, E. Valdinoci, A nonlocal free boundary problem, SIAM Journal on Mathematical Analysis, 47 (2015) pp. 4559--4605.
    Abstract
    We consider a nonlocal free boundary problem built by a fractional Dirichlet norm plus a fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper.

  • R. Rossi, M. Thomas, From an adhesive to a brittle delamination model in thermo-visco-elasticity, ESAIM. Control, Optimisation and Calculus of Variations, 21 (2015) pp. 1--59.
    Abstract
    We address the analysis of a model for brittle delamination of two visco-elastic bodies, bonded along a prescribed surface. The model also encompasses thermal effects in the bulk. The related PDE system for the displacements, the absolute temperature, and the delamination variable has a highly nonlinear character. On the contact surface, it features frictionless Signorini conditions and a nonconvex, brittle constraint acting as a transmission condition for the displacements. We prove the existence of (weak/energetic) solutions to the associated Cauchy problem, by approximating it in two steps with suitably regularized problems. We perform the two consecutive passages to the limit via refined variational convergence techniques.

  • R. Servadei, E. Valdinoci, The Brezis--Nirenberg result for the fractional Laplacian, Transactions of the American Mathematical Society, 367 (2015) pp. 67--102.

  • P. Auscher, N. Badr, R. Haller-Dintelmann, J. Rehberg, The square root problem for second order, divergence form operators with mixed boundary condition on $L^p$, Journal of Evolution Equations, 15 (2015) pp. 165--208.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 26 (2015) pp. 93--108.
    Abstract
    In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

  • P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the pure Cahn--Hilliard equation with dynamic boundary conditions, Advances in Nonlinear Analysis, 4 (2015) pp. 311--325.
    Abstract
    A boundary control problem for the pure Cahn--Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved.

  • P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Optimal boundary control of a viscous Cahn--Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM Journal on Control and Optimization, 53 (2015) pp. 2696--2721.
    Abstract
    In this paper, we investigate optimal boundary control problems for Cahn--Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace--Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see Appl. Math. Optim., 2014) to the (simpler) Allen--Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called ``deep quench limit''. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

  • P. Colli, M.H. Farshbaf Shaker, G. Gilardi, J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn--Hilliard equation with dynamic boundary conditions, Annals of the Academy of Romanian Scientists. Mathematics and its Applications., 7 (2015) pp. 41--66.
    Abstract
    In this paper we establish second-order sufficient optimality conditions for a boundary control problem that has been introduced and studied by three of the authors in the preprint arXiv:1407.3916. This control problem regards the viscous Cahn--Hilliard equation with possibly singular potentials and dynamic boundary conditions.

  • P. Colli, M.H. Farshbaf Shaker, J. Sprekels, A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 71 (2015) pp. 1--24.
    Abstract
    In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors for the case of (differentiable) logarithmic potentials and perform a so-called ``deep quench limit''. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

  • J. Dávila, M. Del Pino, S. Dipierro, E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Analysis & PDE, 8 (2015) pp. 1165--1235.
    Abstract
    For a smooth, bounded Euclidean domain, we consider a nonlocal Schrödinger equation with zero Dirichlet datum. We construct a family of solutions that concentrate at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function in the expanding domain.

  • M. Egert, R. Haller-Dintelmann, J. Rehberg, Hardy's inequality for functions vanishing on a part of the boundary, Potential Analysis, 43 (2015) pp. 49--78.
    Abstract
    We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.

  • M.M. Fall, F. Mahmoudi, E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015) pp. 1937--1961.
    Abstract
    We consider here solutions of the nonlinear fractional Schrödinger equation. We show that concentration points must be critical points for the potential. We also prove that, if the potential is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point. In addition, if the potential is radial, then the minimizer is unique.

  • E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball--Majumdar free energy, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica Ü. Dini", Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 194 (2015) pp. 1269--1299.
    Abstract
    In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $QQ$, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $vt$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in $QQ$ and degenerate in $vt$. To treat it a careful analysis of the properties of $f$, particularly of its blow-up rate, is carried out.

  • A. Fiscella, R. Servadei, E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Mathematical Methods in the Applied Sciences, 38 (2015) pp. 3551--3563.
    Abstract
    In this paper we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation and it is obtained using variational and topological methods. We treat both the nonresonant case and the resonant one.

  • D.A. Gomes, S. Patrizi, Obstacle mean-field game problem, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 17 (2015) pp. 55--68.
    Abstract
    In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.

  • F. Punzo, E. Valdinoci, Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations, Journal of Differential Equations, 258 (2015) pp. 555--587.

  • T. Roubíček, M. Thomas, Ch. Panagiotopoulos, Stress-driven local-solution approach to quasistatic brittle delamination, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015) pp. 645--663.
    Abstract
    A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli.

  • A.F.M. TER Elst, J. Rehberg, Hölder estimates for second-order operators with mixed boundary conditions, Advances in Differential Equations, 20 (2015) pp. 299--360.
    Abstract
    In this paper we investigate linear elliptic, second-order boundary value problems with mixed boundary conditions. Assuming only boundedness/ellipticity on the coefficient function and very mild conditions on the geometry of the domain -- including a very weak compatibility condition between the Dirichlet boundary part and its complement -- we prove first Hölder continuity of the solution. Secondly, Gaussian Hölder estimates for the corresponding heat kernel are derived. The essential instruments are De Giorgi and Morrey-Campanato estimates.

  • M. Thomas, Uniform Poincaré--Sobolev and relative isoperimetric inequalities for classes of domains, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 2741--2761.
    Abstract
    The aim of this paper is to prove an isoperimetric inequality relative to a d-dimensional, bounded, convex domain &Omega intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius r>0 and the position y∈cl(&Omega) of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension d, the diameter of the domain and the integrability exponent p∈[1,d).

  • N. Ahmed, G. Matthies, Higher order continuous Galerkin--Petrov time stepping schemes for transient convection-diffusion-reaction equations, ESAIM: Mathematical Modelling and Numerical Analysis, 49 (2015) pp. 1429--1450.
    Abstract
    We present the analysis for the higher order continuous Galerkin--Petrov (cGP) time discretization schemes in combination with the one-level local projection stabilization in space applied to time-dependent convection-diffusion-reaction problems. Optimal a-priori error estimates will be proved. Numerical studies support the theoretical results. Furthermore, a numerical comparison between continuous Galerkin--Petrov and discontinuous Galerkin time discretization schemes will be given.

  • W. Dreyer, R. Huth, A. Mielke, J. Rehberg, M. Winkler, Global existence for a nonlocal and nonlinear Fokker--Planck equation, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 66 (2015) pp. 293--315.
    Abstract
    We consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state.

  • M.H. Farshbaf Shaker, Ch. Heinemann, A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media, Mathematical Models & Methods in Applied Sciences, 25 (2015) pp. 2749--2793.
    Abstract
    In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility as well as boundedness of the phase field variable which results in a doubly constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model . To this end, we prove existence of minimizers and study a family of ``local'' approximations via adapted cost functionals.

  • M.H. Farshbaf Shaker, A relaxation approach to vector-valued Allen--Cahn MPEC problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 72 (2015) pp. 325--351.

  • H. Hanke, D. Knees, Homogenization of elliptic systems with non-periodic, state dependent coefficients, Asymptotic Analysis, 92 (2015) pp. 203--234.
    Abstract
    In this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided.

  • CH. Heinemann, Ch. Kraus, A degenerating Cahn--Hilliard system coupled with complete damage processes, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015) pp. 388--403.
    Abstract
    Complete damage in elastic solids appears when the material looses all its integrity due to high exposure. In the case of alloys, the situation is quite involved since spinodal decomposition and coarsening also occur at sufficiently low temperatures which may lead locally to high stress peaks. Experimental observations on solder alloys reveal void and crack growth especially at phase boundaries. In this work, we investigate analytically a degenerating PDE system with a time-depending domain for phase separation and complete damage processes under time-varying Dirichlet boundary conditions. The evolution of the system is described by a degenerating parabolic differential equation of fourth order for the concentration, a doubly nonlinear differential inclusion for the damage process and a degenerating quasi-static balance equation for the displacement field. All these equations are strongly nonlinearly coupled. Because of the doubly degenerating character and the doubly nonlinear differential inclusion we are confronted with introducing a suitable notion of weak solutions. We choose a notion of weak solutions which consists of weak formulations of the diffusion equation and the momentum balance, a one-sided variational inequality for the damage function and an energy estimate. For the introduced degenerating system, we prove existence of weak solutions in an $SBV$-framework. The existence result is based on an approximation system, where the elliptic degeneracy of the displacement field and the parabolic degeneracy of the concentration are eliminated. In the framework of phase separation and damage, this means that the approximation system allows only for partial damage and a non-degenerating mobility tensor. For the approximation system, existence results are established. Then, a passage to the limit shows existence of weak solutions of the degenerating system.

  • CH. Heinemann, Ch. Kraus, Complete damage in linear elastic materials -- Modeling, weak formulation and existence results, Calculus of Variations and Partial Differential Equations, 54 (2015) pp. 217--250.
    Abstract
    We introduce a complete damage model with a time-depending domain for linear-elastically stressed solids under time-varying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasi-static balance equation for the displacement field. For the introduced complete damage model, we propose a classical formulation and a corresponding suitable weak formulation in an $SBV$-framework. We show that the classical differential inclusion can be regained from the notion of weak solutions under certain regularity assumptions. The main aim of this work is to prove local-in-time existence and global-in-time existence in some weaker sense for the introduced model. In contrast to incomplete damage theories, the material can be exposed to damage in the proposed model in such a way that the elastic behavior may break down on the damaged parts of the material, i.e. we loose coercivity properties of the free energy. This leads to several mathematical difficulties. For instance, it might occur that not completely damaged material regions are isolated from the Dirichlet boundary. In this case, the deformation field cannot be controlled in the transition from incomplete to complete damage. To tackle this problem, we consider the evolution process on a time-depending domain. In this context, two major challenges arise: Firstly, the time-dependent domain approach leads to jumps in the energy which have to be accounted for in the energy inequality of the notion of weak solutions. To handle this problem, several energy estimates are established by $Gamma$-convergence techniques. Secondly, the time-depending domain might have bad smoothness properties such that Korn's inequality cannot be applied. To this end, a covering result for such sets with smooth compactly embedded domains has been shown.

  • CH. Heinemann, Ch. Kraus, Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 2565--2590.
    Abstract
    In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first generalize the notion of weak solution introduced in WIAS 1520. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.

  • CH. Heinemann, Ch. Kraus, Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains, SIAM Journal on Mathematical Analysis, 47 (2015) pp. 2044--2073.
    Abstract
    The aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertial effects. To this end, we first present a suitable weak formulation in order to deal with such evolution inclusions. Then, existence of weak solutions is proven by utilizing time-discretization, $H^2$--regularization and variational techniques.

  • CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015) pp. 4587--4612.
    Abstract
    In this paper we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging.

  • A. Mielke, J. Naumann, Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence, Comptes Rendus Mathematique. Academie des Sciences. Paris, 353 (2015) pp. 321--326.
    Abstract
    We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in ℝ3. This model consists in a Navier-Stokes type system for the mean flow u and two further partial differential equations: an equation for the frequency ω and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder Ω x ]0,T[ (Ω ⊂ ℝ3 cube, 0 < T < +∞) under spatial periodic boundary conditions on ∂Ω x ]0,T[ and initial conditions in Ω x {0}. We present an existence result for a weak solution {u, ω, k} to the problem under consideration, with ω, k obeying the inequalities formula1 and formula2.

  • A. Mielke, Deriving amplitude equations via evolutionary Gamma convergence, Discrete and Continuous Dynamical Systems, 35 (2015) pp. 2679--2700.
    Abstract
    We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary Gamma convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show Gamma convergence of the associated energies in suitable function spaces. The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in L2, while for the case of a quadratic nonlinearity we need to impose weak convergence in H1. However, we do not need wellpreparedness of the initial conditions.

  • A. Mielke, J. Haskovec, P.A. Markowich, On uniform decay of the entropy for reaction-diffusion systems, Journal of Dynamics and Differential Equations, 27 (2015) pp. 897--928.
    Abstract
    In this work we derive entropy decay estimates for a class of nonlinear reaction-diffusion systems modeling reversible chemical reactions under the assumption of detailed balance. In particular, we provide explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the log-Sobolev inequality and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: we allow for vanishing diffusion constants in some chemical components, and we consider different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

  • E. Rocca, R. Rossi, A degenerating PDE system for phase transitions and damage, Mathematical Models & Methods in Applied Sciences, 24 (2014) pp. 1265--1341.

  • S. Heinz, On the structure of the quasiconvex hull in planar elasticity, Calculus of Variations and Partial Differential Equations, 50 (2014) pp. 481--489.
    Abstract
    Let K and L be compact sets of real 2x2 matrices with positive determinant. Suppose that both sets are frame invariant, meaning invariant under the left action of the special orthogonal group. Then we give an algebraic characterization for K and L to be incompatible for homogeneous gradient Young measures. This result permits a simplified characterization of the quasiconvex hull and the rank-one convex hull in planar elasticity.

  • S. Jachalski, G. Kitavtsev, R. Taranets, Weak solutions to lubrication systems describing the evolution of bilayer thin films, Communications in Mathematical Sciences, 12 (2014) pp. 527--544.
    Abstract
    The existence of global nonnegative weak solutions is proved for coupled one-dimen- sional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

  • B. Barrios, I. Peral, F. Soria, E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Archive for Rational Mechanics and Analysis, 213 (2014) pp. 629--650.

  • A. Cesaroni, M. Novaga, E. Valdinoci, A symmetry result for the Ornstein--Uhlenbeck operator, Discrete and Continuous Dynamical Systems, 34 (2014) pp. 2451--2467.

  • R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 144 (2014) pp. 831--855.

  • N. Abatangelo, E. Valdinoci, A notion of nonlocal curvature, Numerical Functional Analysis and Optimization. An International Journal, 35 (2014) pp. 793--815.

  • P. Colli, G. Gilardi, P. Krejčí, P. Podio-Guidugli, J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous Cahn--Hilliard system, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014) pp. 1061--1087.
    Abstract
    In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.

  • P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, Mathematical Methods in the Applied Sciences, 37 (2014) pp. 1318--1324.
    Abstract
    The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.

  • P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evolution Equations and Control Theory, 3 (2014) pp. 257--275.
    Abstract
    We are concerned with a nonstandard phase field model of Cahn--Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.

  • P. Colli, G. Gilardi, J. Sprekels, On the Cahn--Hilliard equation with dynamic boundary conditions and a dominating boundary potential, Journal of Mathematical Analysis and Applications, 419 (2014) pp. 972--994.
    Abstract
    The Cahn--Hilliard and viscous Cahn--Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.

  • M. Cozzi, A. Farina, E. Valdinoci, Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations, Communications in Mathematical Physics, 331 (2014) pp. 189--214.

  • M.M. Fall, E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of (-Delta) su+u=up in RN when s is close to 1, Communications in Mathematical Physics, 329 (2014) pp. 383--404.

  • A. Farina, E. Valdinoci, Gradient bounds for anisotropic partial differential equations, Calculus of Variations and Partial Differential Equations, 49 (2014) pp. 923--936.

  • A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 94 (2014) pp. 156--170.

  • A. Gloria, S. Neukamm, F. Otto, An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014) pp. 325--346.
    Abstract
    We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author.

  • R. Haller-Dintelmann, W. Höppner, H.-Chr. Kaiser, J. Rehberg, G. Ziegler, Optimal elliptic Sobolev regularity near three-dimensional, multi-material Neumann vertices, Functional Analysis and its Applications, 48 (2014) pp. 208--222.
    Abstract
    We investigate optimal elliptic regularity (within the scale of Sobolev spaces) of anisotropic div-grad operators in three dimensions at a multi-material vertex on the Neumann boundary part of the polyhedral spatial domain. The gradient of a solution to the corresponding elliptic PDE (in a neighbourhood of the vertex) is integrable to an index greater than three.

  • P. Hornung, S. Neukamm, I. Velcic, Derivation of a homogenized nonlinear plate theory from 3D elasticity, Calculus of Variations and Partial Differential Equations, 51 (2014) pp. 677--699.

  • S. Melchionna, E. Rocca, On a nonlocal Cahn--Hilliard equation with a reaction term, Advances in Mathematical Sciences and Applications, 24 (2014) pp. 461--497.
    Abstract
    We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn- Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in u reaction term g(x, t, u).

  • A. Miranville, E. Rocca, G. Schimperna, A. Segatti, The Penrose--Fife phase-field model with coupled dynamic boundary conditions, Discrete and Continuous Dynamical Systems, 34 (2014) pp. 4259--4290.

  • O. Savin, E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, Journal de Mathématiques Pures et Appliquées, 101 (2014) pp. 1--26.

  • A.F.M. TER Elst, M. Meyries, J. Rehberg, Parabolic equations with dynamical boundary conditions and source terms on interfaces, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica Ü. Dini", Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 193 (2014) pp. 1295--1318.
    Abstract
    We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable Lp-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.

  • D.A. Gomes, S. Patrizi, V. Voskanyan, On the existence of classical solutions for stationary extended mean field games, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 99 (2014) pp. 49--79.

  • A. Mielke, S. Reichelt, M. Thomas, Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Networks Heterogeneous Media, 9 (2014) pp. 353--382.
    Abstract
    We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

  • A. Mielke, Ch. Ortner, Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM Journal on Mathematical Analysis, 46 (2014) pp. 1317--1347.
    Abstract
    We formulate quasistatic nonlinear finite-strain viscoelasticity of rate-type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame-indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous in some case in a specific case.

  • A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014) pp. 1293--1325.
    Abstract
    Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(zt,żt)=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

  • A. Mielke, S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, XIII (2014) pp. 67--135.
    Abstract
    We consider quasilinear parabolic systems with a nonsmooth rate-independent dissipation term in the limit of very slow loading rates, or equivalently with fixed loading and vanishing viscosity $varepsilon>0$. Because for nonconvex energies the solutions will develop jumps, we consider the vanishing-viscosity limit for the graphs of the solutions in the extended state space in arclength parametrization, where the norm associated with the viscosity is used to keep the subdifferential structure of the problem. A crucial point in the analysis are new a priori estimates that are rate independent and that allows us to show that the total length of the graph remains bounded in the vanishing-viscosity limit. To derive these estimates we combine parabolic regularity estimates with ideas from rate-independent systems.

  • S. Neukamm, H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calculus of Variations and Partial Differential Equations, (published online on Sept. 14, 2014) pp. , DOI 10.1007/s00526-014-0765-2 .
    Abstract
    We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

  • P.-É. Druet, Global Lipschitz continuity for elliptic transmission problems with a boundary intersecting interface, Mathematica Bohemica, 138 (2013) pp. 185--224.
    Abstract
    We investigate the regularity of the weak solution to elliptic transmission problems that involve two layered anisotropic materials separated by a boundary intersecting interface. Under a compatibility condition for the angle of contact of the two surfaces and the boundary data, we prove the existence of square-integrable second derivatives, and the global Lipschitz continuity of the solution. We show that the second weak derivatives remain integrable to a certain power less than two if the compatibility condition is violated.

  • CH. Heinemann, Ch. Kraus, Existence results for diffuse interface models describing phase separation and damage, European Journal of Applied Mathematics, 24 (2013) pp. 179--211.
    Abstract
    In this paper we analytically investigate Cahn-Hilliard and Allen-Cahn systems which are coupled with elasticity and uni-directional damage processes. We are interested in the case where the free energy contains logarithmic terms of the chemical concentration variable and quadratic terms of the gradient of the damage variable. For elastic Cahn-Hilliard and Allen-Cahn systems coupled with uni-directional damage processes, an appropriate notion of weak solutions is presented as well as an existence result based on certain regularization methods and an higher integrability result for the strain.

  • K. Götze, Strong solutions for the interaction of a rigid body and a viscoelastic fluid, Journal of Mathematical Fluid Mechanics, 15 (2013) pp. 663--688.
    Abstract
    We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.

  • M. Liero, U. Stefanelli, A new minimum principle for Lagrangian mechanics, Journal of Nonlinear Science, 23 (2013) pp. 179--204.
    Abstract
    We present a novel variational approach to Lagrangian mechanics based on elliptic regularization with respect to time. A class of parameter-dependent global-in-time minimization problems is presented and the convergence of the respective minimizers to the solution of the system of Lagrange's equations is ascertained. Moreover, we extend this perspective to mixed dissipative/nondissipative situations, present a finite time-horizon version of this approach, and provide related Gamma-convergence results. Finally, some discussion on corresponding time-discrete versions of the principle is presented.

  • M. Liero, U. Stefanelli, Weighted Inertia-Dissipation-Energy functionals for semilinear equations, Bollettino della Unione Matematica Italiana. Serie 9, VI (2013) pp. 1--27.

  • M. Liero, A. Mielke, Gradient structures and geodesic convexity for reaction-diffusion systems, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013) pp. 20120346/1--20120346/28.
    Abstract
    We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambda-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.

  • M. Liero, Passing from bulk to bulk/surface evolution in the Allen--Cahn equation, NoDEA. Nonlinear Differential Equations and Applications, 20 (2013) pp. 919--942.
    Abstract
    In this paper we formulate a boundary layer approximation for an Allen-Cahn-type equation involving a small parameter $eps$. Here, $eps$ is related to the thickness of the boundary layer and we are interested in the limit when $eps$ tends to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-gradient flow with the corresponding Allen-Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.

  • S. Neukamm, I. Velcic, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 23 (2013) pp. 2701--2748.
    Abstract
    We rigorously derive a homogenized von-Kármán plate theory as a Gamma-limit from nonlinear three-dimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, three-dimensional plate with spatially periodic material properties. The functional features two small length scales: the period $epsilon$ of the elastic composite material, and the thickness h of the slender plate. We study the behavior as $epsilon$ and h simultaneously converge to zero in the von-Kármán scaling regime. The obtained limit is a homogenized von-Kármán plate model. Its effective material properties are determined by a relaxation formula that exposes a non-trivial coupling of the behavior of the out-of-plane displacement with the oscillatory behavior in the in-plane directions. In particular, the homogenized coefficients depend on the relative scaling between h and $epsilon$, and different values arise for h<<$epsilon$, $epsilon$   h and $epsilon$ << h.

  • A. Fiaschi, D. Knees, S. Reichelt, Global higher integrability of minimizers of variational problems with mixed boundary conditions, Journal of Mathematical Analysis and Applications, 401 (2013) pp. 269--288.
    Abstract
    We consider integral functionals with densities of p-growth, with respect to gradients, on a Lipschitz domain with mixed boundary conditions. The aim of this paper is to prove that, under uniform estimates within certain classes of p-growth and coercivity assumptions on the density, the minimizers are of higher integrability order, meaning that they belong to the space of first order Sobolev functions with an integrability of order p+ε for a uniform ε >0. The results are applied to a model describing damage evolution in a nonlinear elastic body and to a model for shape memory alloys.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, An asymptotic analysis for a nonstandard Cahn--Hilliard system with viscosity, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 353--368.
    Abstract
    This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $rho$ and the chemical potential $mu$; each equation includes a viscosity term -- respectively, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- with $varepsilon$ and $delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(varepsilon,delta)-$solutions. Here we discuss the asymptotic limit of the system as $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity, Journal of Differential Equations, 254 (2013) pp. 4217--4244.
    Abstract
    Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.

  • M.H. Duong, V. Laschos, M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013) pp. 1166--1188.

  • M. Geissert, K. Götze, M. Hieber, $L^p$-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids, Transactions of the American Mathematical Society, 365 (2013) pp. 1393--1439.
    Abstract
    Consider the system of equations describing the motion of a rigid body immersed in a viscous, incompressible fluid of Newtonian or generalized Newtonian type. The class of fluids considered includes in particular shear-thinning or shear-thickening fluids of power-law type of exponent $ dgeq 1$. We develop a method to prove that this system admits a unique, local, strong solution in the $ L^p$-setting. The approach presented in the case of generalized Newtonian fluids is based on the theory of quasi-linear evolution equations and requires that the exponent $ p$ satisfies the condition $ p>5$.

  • M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the Fokker--Planck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013) pp. 1350017/1--1350017/43.

  • A. Glitzky, A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 64 (2013) pp. 29--52.
    Abstract
    We derive gradient-flow formulations for systems describing drift-diffusion processes of a finite number of species which undergo mass-action type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electro-reaction-diffusion systems with active interfaces permitting drift-diffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the self-consistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models.

  • D. Knees, R. Rossi, Ch. Zanini, A vanishing viscosity approach to a rate-independent damage model, Mathematical Models & Methods in Applied Sciences, 23 (2013) pp. 565--616.
    Abstract
    We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

  • D. Knees, A. Schröder, Computational aspects of quasi-static crack propagation, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 63--99.
    Abstract
    The focus of this note lies on the numerical analysis of models describing the propagation of a single crack in a linearly elastic material. The evolution of the crack is modeled as a rate-independent process based on the Griffith criterion. We follow two different approaches for setting up mathematically well defined models: the global energetic approach and an approach based on a viscous regularization. We prove the convergence of solutions of fully discretized models (i.e. with respect to time and space) and derive relations between the discretization parameters (mesh size, time step size, viscosity parameter, crack increment) which guarantee the convergence of the schemes. Further, convergence rates are provided for the approximation of energy release rates by certain discrete energy release rates. Thereby we discuss both, models with self-contact conditions on the crack faces as well as models with pure Neumann conditions on the crack faces. The convergence proofs rely on regularity estimates for the elastic fields close to the crack tip and local and global finite element error estimates. Finally the theoretical results are illustrated with some numerical calculations.

  • M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 235--255.
    Abstract
    An existence result for energetic solutions of rate-independent damage processes is established. We consider a body consisting of a physically linearly elastic material undergoing infinitesimally small deformations and partial damage. In [ThomasMielke10DamageZAMM] an existence result in the small strain setting was obtained under the assumption that the damage variable z satisfies z∈ W1,r(Ω) with r∈(1,∞) for Ω⊂Rd. We now cover the case r=1. The lack of compactness in W1,1(Ω) requires to do the analysis in BV(Ω). This setting allows it to consider damage variables with values in 0,1. We show that such a brittle damage model is obtained as the Γ-limit of functionals of Modica-Mortola type.

  • D. Hömberg, K. Krumbiegel, J. Rehberg, Optimal control of a parabolic equation with dynamic boundary condition, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 67 (2013) pp. 3--31.
    Abstract
    We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the Robin boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an $L^p$ function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.

  • A. Mielke, R. Rossi, G. Savaré, Nonsmooth analysis of doubly nonlinear evolution equations, Calculus of Variations and Partial Differential Equations, 46 (2013) pp. 253--310.
    Abstract
    In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.

  • A. Mielke, E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model, Mathematical Models & Methods in Applied Sciences, 23 (2013) pp. 873--916.
    Abstract
    We consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated form the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

  • A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calculus of Variations and Partial Differential Equations, 48 (2013) pp. 1-31.
    Abstract
    We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some more general nonlinear reaction systems

  • A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 479--499.
    Abstract
    We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system (X,Φ,K) defining the evolution $dot U$= - K(U) DΦ(U). Here Φ is the driving functional, while the Onsager operator K(U) is symmetric and positive semidefinite. If the inverse G=K-1 exists, the triple (X,Φ,G) defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential Ψ*(U, Ξ)= ½ ⟨Ξ K(U) Ξ⟩. Then, the two functionals Φ and Ψ* can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.

  • P.-E. Druet, The classical solvability of the contact angle problem for generalized equations of mean curvature type, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 69 (2012) pp. 233--258.
    Abstract
    In this paper, mean curvature type equations with general potentials and contact angle boundary conditions are considered. We extend the ideas of Ural'tseva, formulating sharper hypotheses for the existence of a classical solution. Corner stone for these results is a method to estimate quantities on the boundary of the free surface. We moreover provide alternative proofs for the higher-order estimates, and for the existence result.

  • M. Liero, Th. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $Gamma$-convergence, NoDEA. Nonlinear Differential Equations and Applications, 19 (2012) pp. 437--457.
    Abstract
    This paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance.

  • A. Fiaschi, D. Knees, U. Stefanelli, Young measure quasi-static damage evolution, Archive for Rational Mechanics and Analysis, 203 (2012) pp. 415--453.

  • S. Arnrich, A. Mielke, M.A. Peletier, G. Savaré, M. Veneroni, Passing to the limit in a Wasserstein gradient flow: From diffusion to reaction, Calculus of Variations and Partial Differential Equations, 44 (2012) pp. 419--454.
    Abstract
    We study a singular-limit problem arising in the modelling of chemical reactions. At finite $e>0$, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by $1/e$, and in the limit $eto0$, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savaré, and Veneroni, em SIAM Journal on Mathematical Analysis, 42(4):1805--1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular, we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the propety of being a emphcurve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the $e$-problem converge to a solution of the limiting problem.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Continuous dependence for a nonstandard Cahn--Hilliard system with nonlinear atom mobility, Rendiconti del Seminario Matematico. Universita e Politecnico Torino, 70 (2012) pp. 27--52.
    Abstract
    This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice [Podio-Guidugli 2006]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter $rho$ and the chemical potential $mu$. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of argumentation developed in Colli/Gilardi/Podio-Guidugli/Sprekels 2012.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Continuum Mechanics and Thermodynamics, 24 (2012) pp. 437--459.
    Abstract
    We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in [4], on the basis of the theory developed in [15], and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence for a strongly coupled Cahn--Hilliard system with viscosity, Bollettino della Unione Matematica Italiana. Serie 9, 5 (2012) pp. 495--513.
    Abstract
    An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [CGPS11]. Both systems conform to the general theory developed in [Pod06]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $rho$ and the chemical potential $mu$. In the system studied in this note, a phase-field equation in $rho$ fairly more general than in [CGPS11] is coupled with a highly nonlinear diffusion equation for $mu$, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.

  • P. Colli, G. Gilardi, J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan Journal of Mathematics, 80 (2012) pp. 119--149.
    Abstract
    We investigate a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced in Podio-Guidugli (2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in Colli, Gilardi, Podio-Guidugli, and Sprekels (2011a and b) for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.

  • K. Hackl, S. Heinz, A. Mielke, A model for the evolution of laminates in finite-strain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 92 (2012) pp. 888--909.
    Abstract
    We study the time evolution in elastoplasticity within the rate-independent framework of generalized standard materials. Our particular interest is the formation and the evolution of microstructure. Providing models where existence proofs are possible is a challenging task since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with generalized standard materials. More precisely, we shall allow for such microstructure that is given by simple or sequential laminates. We will consider a model for the evolution of these laminates and we will prove a theorem on the existence of solutions to any finite sequence of time-incremental minimization problems. In order to illustrate the mechanical consequences of the theory developed some numerical results, especially dealing with the rotation of laminates, are presented.

  • A.F.M. TER Elst, J. Rehberg, $L^infty$-estimates for divergence operators on bad domains, Analysis and Applications, 10 (2012) pp. 207--214.
    Abstract
    In this paper, we prove $L^infty$-estimates for solutions of divergence operators in case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.

  • G. Aki, J. Daube, W. Dreyer, J. Giesselmann, M. Kränkel, Ch. Kraus, A diffuse interface model for quasi-incompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012) pp. 54--77.
    Abstract
    In this contribution, we investigate a diffuse interface model for quasi-incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn-Hilliard equation numerically by a Local Discontinuous Galerkin scheme.

  • A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM Journal on Mathematical Analysis, 44 (2012) pp. 3874--3900.
    Abstract
    We introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generation-recombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level.

  • D. Knees, A. Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, Mathematical Methods in the Applied Sciences, 35 (2012) pp. 1859--1884.
    Abstract
    A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D) the displacement fields are B 3/2 2,∞ regular. The proof relies on a difference quotient argument for the directions tangential to the crack. In order to obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. Based on Falk's approximation Theorem for variational inequalities, convergence rates for FE-discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results.

  • W. Dreyer, J. Giesselmann, Ch. Kraus, Ch. Rohde, Asymptotic analysis for Korteweg models, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 14 (2012) pp. 105--143.
    Abstract
    This paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.

  • A. Mielke, R. Rossi, G. Savaré, Variational convergence of gradient flows and rate-independent evolutions in metric spaces, Milan Journal of Mathematics, 80 (2012) pp. 381--410.
    Abstract
    We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p>1, when the exponents p converge to 1.

  • A. Mielke, Emergence of rate-independent dissipation from viscous systems with wiggly energies, Continuum Mechanics and Thermodynamics, 24 (2012) pp. 591--606.
    Abstract
    We consider the passage from viscous system to rate-independent system in the limit of vanishing viscosity and for wiggly energies. Our new convergence approach is based on the (R,R*) formulation by De Giorgi, where we pass to the Γ limit in the dissipation functional. The difficulty is that the type of dissipation changes from a quadratic functional to one that is homogeneous of degree 1. The analysis uses the decomposition of the restoring force into a macroscopic part and a fluctuating part, where the latter is handled via homogenization.

  • CH. Heinemann, Ch. Kraus, Existence of weak solutions for Cahn--Hilliard systems coupled with elasticity and damage, Advances in Mathematical Sciences and Applications, 21 (2011) pp. 321--359.
    Abstract
    The Cahn-Hilliard model is a typical phase field approach for describing phase separation and coarsening phenomena in alloys. This model has been generalized to the so-called Cahn-Larché system by combining it with elasticity to capture non-neglecting deformation phenomena, which occurs during phase separation in the material. Evolving microstructures such as phase separation and coarsening processes have a strong influence on damage initiation and propagation in alloys. We develop the existing framework of Cahn-Hilliard and Cahn-Larché systems by coupling these systems with a unidirectional evolution inclusion for an internal variable, describing damage processes. After establishing a weak notion of the corresponding evolutionary system, we prove existence of weak solutions for rate-dependent damage processes under certain growth conditions of the energy functional.

  • P. Colli, P. Krejčí, E. Rocca, J. Sprekels, A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity, Journal of Differential Equations, 251 (2011) pp. 1354--1387.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn--Hilliard system, SIAM Journal on Applied Mathematics, 71 (2011) pp. 1849--1870.
    Abstract
    We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative $omega$-limit set.

  • R. Haller-Dintelmann, H.-Chr. Kaiser, J. Rehberg, Direct computation of elliptic singularities across anisotropic, multi-material edges, Journal of Mathematical Sciences (New York), 172 (2011) pp. 589--622.
    Abstract
    We characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and four-material edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark L--shape problem.

  • K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the Navier--Stokes--Korteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011) pp. 239--254.
    Abstract
    In this contribution we will study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model we will consider the static version of the Navier-Stokes-Korteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations.

  • A. Glitzky, Analysis of electronic models for solar cells including energy resolved defect densities, Mathematical Methods in the Applied Sciences, 34 (2011) pp. 1980--1998.
    Abstract
    We introduce an electronic model for solar cells including energy resolved defect densities. The resulting drift-diffusion model corresponds to a generalized van Roosbroeck system with additional source terms coupled with ODEs containing space and energy as parameters for all defect densities. The system has to be considered in heterostructures and with mixed boundary conditions from device simulation. We give a weak formulation of the problem. If the boundary data and the sources are compatible with thermodynamic equilibrium the free energy along solutions decays monotonously. In other cases it may be increasing, but we estimate its growth. We establish boundedness and uniqueness results and prove the existence of a weak solution. This is done by considering a regularized problem, showing its solvability and the boundedness of its solutions independent of the regularization level.

  • CH. Kraus, The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs--Thomson law, European Journal of Applied Mathematics, 22 (2011) pp. 393--422.
    Abstract
    The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs-Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end approximate solutions are constructed by means of variational functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs-Thomson law in a weak generalized BV-formulation.

  • A. Mielke, U. Stefanelli, Weighted energy-dissipation functionals for gradient flows, ESAIM. Control, Optimisation and Calculus of Variations, 17 (2011) pp. 52--85.
    Abstract
    We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke & Ortiz in ``A class of minimum principles for characterizing the trajectories of dissipative systems''. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from S. Conti and M. Ortiz ``Minimum principles for the trajectories of systems governed by rate problems''.

  • A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011) pp. 1329--1346.
    Abstract
    In recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reaction-diffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature.

  • A. Mielke, Complete-damage evolution based on energies and stresses, Discrete and Continuous Dynamical Systems -- Series S, 4 (2011) pp. 423--439.
    Abstract
    The rate-independent damage model recently developed in Bouchitté, Mielke, Roubíček ``A complete-damage problem at small strains" allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions.

  • A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems, Mathematische Nachrichten, 284 (2011) pp. 2159--2174.
    Abstract
    Our focus are energy estimates for discretized reaction-diffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete Sobolev-Poincaré inequality.

  • J.A. Griepentrog, L. Recke, Local existence, uniqueness, and smooth dependence for nonsmooth quasilinear parabolic problems, Journal of Evolution Equations, 10 (2010) pp. 341--375.
    Abstract
    A general theory on local existence, uniqueness, regularity, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data has been developed. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to the abstract formulations of the initial boundary value problems, has been closed. The main tools are new maximal regularity results of the first author in Sobolev-Morrey spaces, linearization techniques and the Implicit Function Theorem. Typical applications are transport processes of charged particles in semiconductor heterostructures, phase separation processes of nonlocally interacting particles, chemotactic aggregation in heterogeneous environments as well as optimal control by means of quasilinear elliptic and parabolic PDEs with nonsmooth data.

  • M. Thomas, A. Mielke, Damage of nonlinearly elastic materials at small strain --- Existence and regularity results, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010) pp. 88--112.
    Abstract
    In this paper an existence result for energetic solutions of rate-independent damage processes is established and the temporal regularity of the solution is discussed. We consider a body consisting of a physically nonlinearly elastic material undergoing small deformations and partial damage. The present work is a generalization of [Mielke-Roubicek 2006] concerning the properties of the stored elastic energy density as well as the suitable Sobolev space for the damage variable: While previous work assumes that the damage variable z satisfies z ∈ W^1,r (Omega) with r>d for Omega ⊂ R^d, we can handle the case r>1 by a new technique for the construction of joint recovery sequences. Moreover, this work generalizes the temporal regularity results to physically nonlinearly elastic materials by analyzing Lipschitz- and Hölder-continuity of solutions with respect to time.

  • H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified Gibbs--Thomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010) pp. 511--545.
    Abstract
    We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase field energy.

  • J. Giannoulis, A. Mielke, Ch. Sparber, High-frequency averaging in semi-classical Hartree-type equations, Asymptotic Analysis, 70 (2010) pp. 87--100.
    Abstract
    We investigate the asymptotic behavior of solutions to semi-classical Schröodinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.

  • P. Gruber, D. Knees, S. Nesenenko, M. Thomas, Analytical and numerical aspects of time-dependent models with internal variables, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 90 (2010) pp. 861--902.
    Abstract
    In this paper some analytical and numerical aspects of time-dependent models with internal variables are discussed. The focus lies on elasto/visco-plastic models of monotone type arising in the theory of inelastic behavior of materials. This class of problems includes the classical models of elasto-plasticity with hardening and viscous models of the Norton-Hoff type. We discuss the existence theory for different models of monotone type, give an overview on spatial regularity results for solutions to such models and illustrate a numerical solution algorithm at an example. Finally, the relation to the energetic formulation for rate-independent processes is explained and temporal regularity results based on different convexity assumptions are presented.

  • R. Haller-Dintelmann, J. Rehberg, Coercivity for elliptic operators and positivity of solutions on Lipschitz domains, Archiv der Mathematik, 95 (2010) pp. 457--468.
    Abstract
    We show that usual second order operators in divergence form satisfy coercivity on Lipschitz domains if they are either complemented with homogeneous Dirichlet boundary conditions on a set of non-zero boundary measure or if a suitable Robin boundary condition is posed. Moreover, we prove the positivity of solutions in a general, abstract setting, provided that the right hand side is a positive functional. Finally, positive elements from $W^-1,2$ are identified as positive measures.

  • W. Dreyer, Ch. Kraus, On the van der Waals--Cahn--Hilliard phase-field model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010) pp. 1161--1186.
    Abstract
    We study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory.
    For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order.
    The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density.

  • A. Glitzky, K. Gärtner, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, SIAM Journal on Mathematical Analysis, 41 (2010) pp. 2489--2513.
    Abstract
    We study a stationary spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. In 3D we prove the existence and boundedness of steady states. If the Dirichlet conditions are compatible or nearly compatible with thermodynamic equilibrium the solution is unique. The same properties are obtained for a space discretized version of the problem: Using a Scharfetter-Gummel scheme on 3D boundary conforming Delaunay grids we show existence, boundedness and, for small applied voltages, the uniqueness of the discrete solution.

  • D. Hömberg, Ch. Meyer, J. Rehberg, W. Ring, Optimal control for the thermistor problem, SIAM Journal on Control and Optimization, 48 (2010) pp. 3449--3481.
    Abstract
    This paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem.

  • H.-Chr. Kaiser, J. Rehberg, Optimal elliptic regularity at the crossing of a material interface and a Neumann boundary edge, Journal of Mathematical Sciences (New York), 169 (2010) pp. 145--166.
    Abstract
    We investigate optimal elliptic regularity of anisotropic div-grad operators in three dimensions at the crossing of a material interface and an edge of the spatial domain on the Neumann boundary part within the scale of Sobolev spaces.

  • D. Knees, On global spatial regularity and convergence rates for time dependent elasto-plasticity, Mathematical Models & Methods in Applied Sciences, 20 (2010) pp. 1823--1858.

  • A. Mielke, T. Roubíček, J. Zeman, Complete damage in elastic and viscoelastic media and its energetics, Computer Methods in Applied Mechanics and Engineering, 199 (2010) pp. 1242--1253.
    Abstract
    A model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The ``flow rule'' for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.

  • J. Sprekels, H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 72 (2010) pp. 3028--3048.

  • K. Hoke, H.-Chr. Kaiser, J. Rehberg, Analyticity for some operator functions from statistical quantum mechanics, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 10 (2009) pp. 749--771.
    Abstract
    For rather general thermodynamic equilibrium distribution functions the density of a statistical ensemble of quantum mechanical particles depends analytically on the potential in the Schrödinger operator describing the quantum system. A key to the proof is that the resolvent to a power less than one of an elliptic operator with non-smooth coefficients, and mixed Dirichlet/Neumann boundary conditions on a bounded up to three-dimensional Lipschitz domain factorizes over the space of essentially bounded functions.

  • A. Petrov, M. Schatzman, Mathematical results on existence for viscoelastodynamic problems with unilateral constraints, SIAM Journal on Mathematical Analysis, 40 (2009) pp. 1882--1904.
    Abstract
    We study a damped wave equation and the evolution of a Kelvin-Voigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution.

  • G. Bouchitté, A. Mielke, T. Roubíček, A complete-damage problem at small strains, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009) pp. 205--236.
    Abstract
    The complete damage of a linearly-responding material that can thus completely disintegrate is addressed at small strains under time-varying Dirichlet boundary conditions as a rate-independent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $Gamma$-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example.

  • R. Haller-Dintelmann, Ch. Meyer, J. Rehberg, A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 60 (2009) pp. 397--428.
    Abstract
    The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.

  • R. Haller-Dintelmann, J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions, Journal of Differential Equations, 247 (2009) pp. 1354--1396.
    Abstract
    We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.

  • A. Mainik, A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, Journal of Nonlinear Science, 19 (2009) pp. 221--248.
    Abstract
    We provide a global existence result for the time-continuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables.

  • S. Zelik, A. Mielke, Multi-pulse evolution and space-time chaos in dissipative systems, Memoirs of the American Mathematical Society, 198 (2009) pp. 1--97.

  • A. Glitzky, K. Gärtner, Energy estimates for continuous and discretized electro-reaction-diffusion systems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009) pp. 788--805.
    Abstract
    We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations.
    We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.
    The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species.

  • A. Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, Discrete and Continuous Dynamical Systems, 25 (2009) pp. 159--174.
    Abstract
    We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electro--reaction--diffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discrete-time version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model.

  • H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of drift-diffusion equations for semiconductor devices: The 2D case, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71 (2009) pp. 1584--1605.
    Abstract
    We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.

  • H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Monotonicity properties of the quantum mechanical particle density: An elementary proof, Monatshefte fur Mathematik, 158 (2009) pp. 179--185.
    Abstract
    An elementary proof of the anti-monotonicity of the quantum mechanical particle density with respect to the potential in the Hamiltonian is given for a large class of admissible thermodynamic equilibrium distribution functions. In particular the zero temperature case is included.

  • D. Knees, On global spatial regularity in elasto-plasticity with linear hardening, Calculus of Variations and Partial Differential Equations, 36 (2009) pp. 611--625.
    Abstract
    We study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the remaining faces. Under natural smoothness assumptions on the data we obtain u in L((0,T);H3/2-δ(Ω)) for the displacements and z in L((0,T);H1/2-δ(Ω)) for the internal variables. The proof is based on a difference quotient technique and a reflection argument.

  • P. Krejčí, M. Liero, Rate independent Kurzweil processes, Applications of Mathematics, 54 (2009) pp. 117--145.
    Abstract
    The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in $BV$ spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.

  • A. Mielke, F. Rindler, Reverse approximation of energetic solutions to rate-independent processes, NoDEA. Nonlinear Differential Equations and Applications, 16 (2009) pp. 17--40.
    Abstract
    Energetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.

  • A. Mielke, R. Rossi, G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems, 25 (2009) pp. 585--615.
    Abstract
    Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given.

  • A. Mielke, T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009) pp. 399--429.
    Abstract
    A general abstract approximation scheme for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The abstract theory is illustrated on several examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

  • A. Mielke, L. Paoli, A. Petrov, On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys, SIAM Journal on Mathematical Analysis, 41 (2009) pp. 1388--1414.
    Abstract
    This paper deals with a three-dimensional model for thermal stress-induced transformations in shape-memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also space-time discretizations and establish convergence of these approximations.

  • H. Neidhardt, V.A. Zagrebnov, Linear non-autonomous Cauchy problems and evolution semigroups, Advances in Differential Equations, 14 (2009) pp. 289--340.
    Abstract
    The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

  • H. Stephan, Modeling of drift-diffusion systems, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 60 (2009) pp. 33--53.
    Abstract
    We derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, or - in the case of cross diffusion - even constant.

  • P.-É. Druet, Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ with $pge 1$, Mathematical Methods in the Applied Sciences, 32 (2008) pp. 135--166.
    Abstract
    Accurate modeling of heat transfer in high-temperatures situations requires to account for the effect of heat radiation. In complex applications such as Czochralski's method for crystal growth, in which the conduction radiation heat transfer problem couples to an induction heating problem and to the melt flow problem, we hardly can expect from the mathematical theory that the heat sources will be in a better space than L-1. In such situations, the known results on the unique solvability of the heat conduction problem with surface radiation do not apply, since a right-hand side in L-p with p < 6/5 no longer belongs to the dual of the Banach space in which coercivity is obtained. In this paper, we focus on a stationary heat equation with non-local boundary conditions and right-hand side in L-p with p>=1 arbitrary. Essentially, we construct an approximation procedure and, thanks to new coercivity results, we are able to produce energy estimates that involve only the L-p-norm of the heat-sources, and to pass to the limit.

  • J.A. Griepentrog, W. Höppner, H.-Chr. Kaiser, J. Rehberg, A bi-Lipschitz continuous, volume preserving map from the unit ball onto a cube, Note di Matematica, 28 (2008) pp. 185--201.
    Abstract
    We construct two bi-Lipschitz, volume preserving maps from Euclidean space onto itself which map the unit ball onto a cylinder and onto a cube, respectively. Moreover, we characterize invariant sets of these mappings.

  • S. Heinz, Quasiconvex functions can be approximated by quasiconvex polynomials, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008) pp. 795--801.

  • F. Auricchio, A. Mielke, U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Mathematical Methods in the Applied Sciences, 18 (2008) pp. 125--164.
    Abstract
    This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.

  • H. Cornean, K. Hoke, H. Neidhardt, P.N. Racec, J. Rehberg, A Kohn--Sham system at zero temperature, Journal of Physics. A. Mathematical and General, 41 (2008) pp. 385304/1--385304/21.
    Abstract
    An one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.

  • R. Haller-Dintelmann, H.-Chr. Kaiser, J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de Mathématiques Pures et Appliquées, 89 (2008) pp. 25--48.

  • M. Hieber, J. Rehberg, Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains, SIAM Journal on Mathematical Analysis, 40 (2008) pp. 292--305.
    Abstract
    In this paper we investigate quasilinear systems of reaction-diffusion equations with mixed Dirichlet-Neumann bondary conditions on non smooth domains. Using techniques from maximal regularity and heat-kernel estimates we prove existence of a unique solution to systems of this type.

  • R. Rossi, A. Mielke, G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, VII (2008) pp. 97--169.
    Abstract
    This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces.

  • J.C. DE Los Reyes, P. Merino, J. Rehberg, F. Tröltzsch, Optimality conditions for state-constrained PDE control problems with finite-dimensional control space, Control and Cybernetics, 37 (2008) pp. 7--38.

  • A. Glitzky, R. Hünlich, Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains, Mathematische Nachrichten, 281 (2008) pp. 1676--1693.
    Abstract
    We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a $W^1,p$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem.

  • A. Glitzky, Analysis of a spin-polarized drift-diffusion model, Advances in Mathematical Sciences and Applications, 18 (2008) pp. 401--427.
    Abstract
    We introduce a spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. We give a weak formulation of this problem and prove an existence and uniqueness result for the instationary problem. If the boundary data is compatible with thermodynamic equilibrium the free energy along the solution decays monotonously and exponentially to its equilibrium value. In other cases it may be increasing but we estimate its growth. Moreover we give upper and lower estimates for the solution.

  • A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity, 21 (2008) pp. 1989--2009.
    Abstract
    Our focus are electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electro-reaction-diffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.

  • D. Knees, P. Neff, Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity, SIAM Journal on Mathematical Analysis, 40 (2008) pp. 21--43.
    Abstract
    In this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the time-discretization of models from infinitesimal elasto-plasticity. Our main focus lies on an elasto-plastic Cosserat model. More specifically we show that the time discretization renders $H^2$-regularity of the displacement and $H^1$-regularity for the symmetric plastic strain $varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and Prandtl-Reuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elasto-plasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (Legendre-Hadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the mesh-width $h$ in error estimates.

  • D. Knees, A. Mielke, Energy release rate for cracks in finite-strain elasticity, Mathematical Methods in the Applied Sciences, 31 (2008) pp. 501--528.
    Abstract
    Griffith's fracture criterion describes in a quasistatic setting whether or not a pre-existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: If the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we derive rigorously the well-known Griffith formula and the $J$-integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors.

  • D. Knees, A. Mielke, Ch. Zanini, On the inviscid limit of a model for crack propagation, Mathematical Models & Methods in Applied Sciences, 18 (2008) pp. 1529--1569.
    Abstract
    We study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.

  • D. Knees, A. Mielke, On the energy release rate in finite-strain elasticity, Mechanics of Advanced Materials and Structures, 15 (2008) pp. 421--427.

  • D. Knees, Global stress regularity of convex and some nonconvex variational problems, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica Ü. Dini", Firenze; Springer-Verlag, Heidelberg. English, French, German, Italian, English abstracts., 187 (2008) pp. 157--184.

  • A. Mielke, Weak-convergence methods for Hamiltonian multiscale problems, Discrete and Continuous Dynamical Systems, 20 (2008) pp. 53--79.
    Abstract
    We consider Hamiltonian problems depending on a small parameter like in wave equations with rapidly oscillating coefficients or the embedding of an infinite atomic chain into a continuum by letting the atomic distance tend to $0$. For general semilinear Hamiltonian systems we provide abstract convergence results in terms of the existence of a family of joint recovery operators which guarantee that the effective equation is obtained by taking the $Gamma$-limit of the Hamiltonian. The convergence is in the weak sense with respect to the energy norm. Exploiting the well-developed theory of $Gamma$-convergence, we are able to generalize the admissible coefficients for homogenization in the wave equations. Moreover, we treat the passage from a discrete oscillator chain to a wave equation with general $rmL^infty$ coefficients

  • A. Mielke, A. Petrov, J.A.C. Martins, Convergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit, Journal of Mathematical Analysis and Applications, 348 (2008) pp. 1012--1020.
    Abstract
    This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.

  • J.A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev--Morrey spaces, Advances in Differential Equations, 12 (2007) pp. 1031--1078.
    Abstract
    This text is devoted to maximal regularity results for second order parabolic systems on LIPSCHITZ domains of space dimension greater or equal than three with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of SOBOLEV-MORREY spaces for solutions and right hand sides introduced in the first part of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are HOELDER continuous in time and space up to the boundary for a certain range of MORREY exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients.

  • J.A. Griepentrog, Sobolev--Morrey spaces associated with evolution equations, Advances in Differential Equations, 12 (2007) pp. 781--840.
    Abstract
    In this text we introduce new classes of SOBOLEV-MORREY spaces being adequate for the regularity theory of second order parabolic boundary value problems on LIPSCHITZ domains of space dimension greater or equal than three with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, LIPSCHITZ transformation, and reflection. In the second part of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned SOBOLEV-MORREY spaces of solutions and right hand sides.

  • J. Elschner, H.-Chr. Kaiser, J. Rehberg, G. Schmidt, $W^1,q$ regularity results for elliptic transmission problems on heterogeneous polyhedra, Mathematical Models & Methods in Applied Sciences, 17 (2007) pp. 593--615.

  • J. Elschner, J. Rehberg, G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 9 (2007) pp. 233--252.
    Abstract
    We prove an optimal regularity result for elliptic operators $-nabla cdot mu nabla:W^1,q_0 rightarrow W^-1,q$ for a $q>3$ in the case when the coefficient function $mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.

  • A. Glitzky, R. Hünlich, Resolvent estimates in $W^-1,p$ related to strongly coupled linear parabolic systems with coupled nonsmooth capacities, Mathematical Methods in the Applied Sciences, 30 (2007) pp. 2215--2232.
    Abstract
    We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in $W^-1,p$ spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain $L^p$ estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result.

  • A. Mielke, A. Petrov, Thermally driven phase transformation in shape-memory alloys, Advances in Mathematical Sciences and Applications, 17 (2007) pp. 667--685.
    Abstract
    This paper analyzes a model for phase transformation in shape-memory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rate-independent processes. Existence and uniqueness results are proved.

  • A. Mielke, R. Rossi, Existence and uniqueness results for a class of rate-independent hysteresis problems, Mathematical Models & Methods in Applied Sciences, 17 (2007) pp. 81--123.

  • A. Mielke, A. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM Journal on Mathematical Analysis, 39 (2007) pp. 642--668.
    Abstract
    This paper is devoted to the two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using so-called joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals.

  • A. Mielke, S. Zelik, Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in $R^n$, Journal of Dynamics and Differential Equations, 19 (2007) pp. 333--389.

  • A. Mielke, A model for temperature-induced phase transformations in finite-strain elasticity, IMA Journal of Applied Mathematics, 72 (2007) pp. 644--658.

  • F. Schmid, A. Mielke, Existence results for a contact problem with varying friction coefficient and nonlinear forces, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007) pp. 616--631.
    Abstract
    We consider the rate-independent problem of a particle moving in a three - dimensional half space subject to a time-dependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rate-independent systems as developed by Mielke and co-workers. We generalize the time-incremental minimization procedure of Mielke and Rossi for the present situation of a non-associative flow rule.

  • J. Giannoulis, A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 6 (2006) pp. 493--523.

  • M. Kočvara, A. Mielke, T. Roubíček, A rate-independent approach to the delamination problem, Mathematics and Mechanics of Solids, 11 (2006) pp. 423--447.

  • V. Pata, S. Zelik, A remark on the weakly damped wave equation, Communications on Pure and Applied Analysis, 5 (2006) pp. 609--614.

  • V. Pata, S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006) pp. 1495--1506.

  • H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of quasilinear parabolic systems on two dimensional domains, NoDEA. Nonlinear Differential Equations and Applications, 13 (2006) pp. 287-310.

  • D. Knees, Global regularity of the elastic fields of a power-low model on Lipschitz domains, Mathematical Methods in the Applied Sciences, 29 (2006) pp. 1363--1391.

  • A. Mielke, G. Francfort, Existence results for a class of rate-independent material models with nonconvex elastic energies, Journal fur die Reine und Angewandte Mathematik, 595 (2006) pp. 55--91.

  • A. Mielke, T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 16 (2006) pp. 177--209.

  • A. Mielke, Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner--Husimi transforms, Archive for Rational Mechanics and Analysis, 181 (2006) pp. 401--448.

  • M. Baro, H. Neidhardt, J. Rehberg, Current coupling of drift-diffusion models and dissipative Schrödinger--Poisson systems: Dissipative hybrid models, SIAM Journal on Mathematical Analysis, 37 (2005) pp. 941--981.

  • H. Neidhardt, J. Rehberg, Uniqueness for dissipative Schrödinger--Poisson systems, Journal of Mathematical Physics, 46 (2005) pp. 113513/1--113513/28.

  • M. Kružík, A. Mielke, T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAINi, Meccanica. International Journal of the Italian Association of Theoretical and Applied Mechanics, 40 (2005) pp. 389--418.

  • N. Nefedov, M. Radziunas, K.R. Schneider, A. Vasil'eva, Change of the type of contrast structures in parabolic Neumann problems, Computational Mathematics and Mathematical Physics, 45 (2005) pp. 37--51.

  • W. Dreyer, B. Wagner, Sharp-interface model for eutectic alloys. Part I: Concentration dependent surface tension, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 7 (2005) pp. 199--227.

  • A. Glitzky, R. Hünlich, Global existence result for pair diffusion models, SIAM Journal on Mathematical Analysis, 36 (2005) pp. 1200--1225.

  • A. Glitzky, R. Hünlich, Stationary energy models for semiconductor devices with incompletely ionized impurities, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 85 (2005) pp. 778--792.

  • A. Mielke, A.L. Afendikov, Dynamical properties of spatially non-decaying 2D Navier--Stokes flows with Kolmogorov forcing in an infinite strip, Journal of Mathematical Fluid Mechanics, 7 (2005) pp. 51-67.

  • J. Rehberg, Quasilinear parabolic equations in $L^p$, Progress in Nonlinear Differential Equations and their Applications, 64 (2005) pp. 413-419.

  • B. Wagner, An asymptotic approach to second-kind similarity solutions of the modified porous medium equation, Journal of Engineering Mathematics, 53 (2005) pp. 201-220.

  • M. Baro, H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, A quantum transmitting Schrödinger-Poisson system, Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics, 16 (2004) pp. 281--330.

  • M. Baro, H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Dissipative Schrödinger--Poisson systems, Journal of Mathematical Physics, 45 (2004) pp. 21--43.

  • J. Griepentrog, On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach Center Publications, 66 (2004) pp. 153-164.

  • V. Maz'ya, J. Elschner, J. Rehberg, G. Schmidt, Solutions for quasilinear nonsmooth evolution systems in $L^p$, Archive for Rational Mechanics and Analysis, 171 (2004) pp. 219--262.

  • H. Gajewski, I.V. Skrypnik, On unique solvability of nonlocal drift-diffusion-type problems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 56 (2004) pp. 803--830.

  • H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 10 (2004) pp. 315--336.

  • A. Glitzky, W. Merz, Single dopant diffusion in semiconductor technology, Mathematical Methods in the Applied Sciences, 27 (2004) pp. 133--154.

  • A. Glitzky, R. Hünlich, Stationary solutions of two-dimensional heterogeneous energy models with multiple species, Banach Center Publications, 66 (2004) pp. 135-151.

  • A. Glitzky, Electro-reaction-diffusion systems with nonlocal constraints, Mathematische Nachrichten, 277 (2004) pp. 14--46.

  • H. Gajewski, K. Zacharias, On a nonlocal phase separation model, Journal of Mathematical Analysis and Applications, 286 (2003) pp. 11--31.

  • H. Gajewski, I.V. Skrypnik, On the uniqueness of solutions for nonlinear elliptic-parabolic equations, Journal of Evolution Equations, 3 (2003) pp. 247--281.

  • H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 52 (2003) pp. 291--304.

  • M.A. Efendiev, H. Gajewski, S. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity, Advances in Differential Equations, 7 (2002) pp. 1073--1100.

  • G. Albinus, H. Gajewski, R. Hünlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002) pp. 367--383.

  • J.A. Griepentrog, K. Gröger, H.-Chr. Kaiser, J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Mathematische Nachrichten, 241 (2002) pp. 110--120.

  • J.A. Griepentrog, Linear elliptic boundary value problems with non-smooth data: Campanato spaces of functionals, Mathematische Nachrichten, 243 (2002) pp. 19--42.

  • A. Glitzky, R. Hünlich, Global properties of pair diffusion models, Advances in Mathematical Sciences and Applications, 11 (2001) pp. 293--321.

  • J.A. Griepentrog, H.-Chr. Kaiser, J. Rehberg, Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on $Lsp p$, Advances in Mathematical Sciences and Applications, 11 (2001) pp. 87--112.

  • W. Merz, A. Glitzky, R. Hünlich, K. Pulverer, Strong solutions for pair diffusion models in homogeneous semiconductors, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 2 (2001) pp. 541-567.

  • I.V. Skrypnik, H. Gajewski, On the uniqueness of solution to nonlinear elliptic problem (in Ukrainian), Dopovidi Natsionalnoi Akademii Nauk Ukraini. Matematika. Prirodoznavstvo. Tekhnichni Nauki, (2001) pp. 28--32.

  • J.A. Griepentrog, L. Recke, Linear elliptic boundary value problems with non-smooth data: Normal solvability on Sobolev-Campanato spaces, Mathematische Nachrichten, 225 (2001) pp. 39--74.

  • A. Glitzky, R. Hünlich, Electro-reaction-diffusion systems including cluster reactions of higher order, Mathematische Nachrichten, 216 (2000) pp. 95--118.

  • W. Dreyer, W.H. Müller, A study of the coarsening in tin/lead solders, International Journal of Solids and Structures, 37 (2000) pp. 3841--3871.

  • H.-Chr. Kaiser, J. Rehberg, About a stationary Schrödinger-Poisson system with Kohn-Sham potential in a bounded two- or three-dimensional domain, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 41 (2000) pp. 33--72.

  • J.A. Griepentrog, An application of the Implicit Function Theorem to an energy model of the semiconductor theory, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 79 (1999) pp. 43--51.
    Abstract
    In this paper we deal with a mathematical model for the description of heat conduction and carrier transport in semiconductor heterostructures. We solve a coupled system of nonlinear elliptic differential equations consisting of the heat equation with Joule heating as a source, the Poisson equation for the electric field an drift-diffusion equations with temperature dependent coefficients describing the charge and current conservation, subject to general thermal and electrical boundary conditions. We prove the existence and uniqueness of Holder continuous weak solutions near thermodynamic equilibria points using the Implicit Function Theorem. To show the differentiability of maps corresponding to the weak formulation of the problem we use regularity results from the theory of nonsmooth linear elliptic boundary value problems in Sobolev-Campanato spaces.

  Contributions to Collected Editions

  • G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics, in: MURPHYS-HSFS-2014: 7th International Workshop on MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012009/1--012009/20.
    Abstract
    This note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from [Roubíček M2AS'09, SIAM'10] with the methods from Lazzaroni/Rossi/Thomas/Toader [WIAS Preprint 2025]. The present analysis encompasses, differently from [Roubíček SIAM'10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [WIAS Preprint 2025], a nonconstant heat capacity and a time-dependent Dirichlet loading.

  • M. Thomas, E. Bonetti, E. Rocca, R. Rossi, A rate-independent gradient system in damage coupled with plasticity via structured strains, in: Gradient Flows: From Theory to Application, B. Düring, C.-B. Schönlieb, M.-Th. Wolfram, eds., 54 of ESAIM Proceedings and Surveys, EDP Sciences, 2016, pp. 54--69.
    Abstract
    This contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni, including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers.

  • A. Mielke, R. Rossi, G. Savaré, Balanced-Viscosity solutions for multi-rate systems, in: MURPHYS-HSFS-2014: 7th MUlti-Rate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and Slow-Fast Systems (HSFS), O. Klein, M. Dimian, P. Gurevich, D. Knees, D. Rachinskii, S. Tikhomirov, eds., 727 of Journal of Physics: Conference Series, IOP Publishing, 2016, pp. 012010/1--012010/26.
    Abstract
    Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients εα and ε, where 0<ε<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as ε tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.

  • A. Mielke, Deriving effective models for multiscale systems via evolutionary $Gamma$-convergence, in: Control of Self-Organizing Nonlinear Systems, E. Schöll, S. Klapp, P. Hövel, eds., Understanding Complex Systems, Springer International Publishing AG Switzerland, Cham, 2016, pp. 235--251.

  • A. Mielke, Evolutionary relaxation of a two-phase model, in: Scales in Plasticity, Mini-Workshop, November 8--14, 2015, G.A. Francfort, S. Luckhaus, eds., 12 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2015, pp. 3027--3030.

  • A. Mielke, On thermodynamical couplings of quantum mechanics and macroscopic systems, in: Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 331--348.
    Abstract
    Pure quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use the GENERIC framework to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition:
    formel
    We give three applications of the theory. First, we consider a finite-dimensional quantum system that is coupled to a finite number of simple heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a one-dimensional Schrödinger operator connected to a one-dimensional heat equation on the left and on the right. Finally, we consider thermo-opto-electronics, where the Maxwell-Bloch system of optics is coupled to the energy-drift-diffusion system for semiconductor electronics.

  • A. Fiebach, A. Glitzky, Uniform estimate of the relative free energy by the dissipation rate for finite volume discretized reaction-diffusion systems, in: Finite Volumes for Complex Applications VII -- Methods and Theoretical Aspects -- FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., 77 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 275--283.
    Abstract
    We prove a uniform Poincare-like estimate of the relative free energy by the dissipation rate for implicit Euler, finite volume discretized reaction-diffusion systems. This result is proven indirectly and ensures the exponential decay of the relative free energy with a unified decay rate for admissible finite volume meshes.

  • A. Glitzky, A. Mielke, L. Recke, M. Wolfrum, S. Yanchuk, D2 -- Mathematics for optoelectronic devices, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 243--256.

  • D. Knees, R. Kornhuber, Ch. Kraus, A. Mielke, J. Sprekels, C3 -- Phase transformation and separation in solids, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 189--203.

  • K. Götze, Free fall of a rigid body in a viscoelastic fluid, in: Geophysical Fluid Dynamics, Workshop, February 18--22, 2013, 10 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2013, pp. 554--556.

  • A. Mielke, Gradient structures and dissipation distances for reaction-diffusion systems, in: Material Theory, Workshop, Dezember 16--20, 2013, A. Desimone, S. Luckhaus, L. Truskinovsky, eds., 10 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2013, pp. 3455--3458.

  • K. Götze, Maximal $L^p$-regularity of a 2D fluid-solid interaction problem, in: Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, W. Arendt, J.A. Ball, J. Behrndt, K.-H. Förster, V. Mehrmann, C. Trunk, eds., 221 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2012, pp. 373--384.
    Abstract
    We study a coupled system of equations which appears as a suitable linearisation of the model for the free motion of a rigid body in a Newtonian fluid in two space dimensions. For this problem, we show maximal Lp-regularity estimates.

  • A. Mielke, Multiscale gradient systems and their amplitude equations, in: Dynamics of Pattern, Workshop, Dezember 16--22, 2012, 9 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2012, pp. 3588--3591.

  • R. Haller-Dintelmann, J. Rehberg, Maximal parabolic regularity for divergence operators on distribution spaces, in: Parabolic Problems: The Herbert Amann Festschrift, J. Escher, P. Guidotti, M. Hieber, P. Mucha, J.W. Pruess, Y. Shibata, G. Simonett, Ch. Walker, W. Zajaczkowski, eds., 80 of Progress in Nonlinear Differential Equations and Their Applications, Springer, Basel, 2011, pp. 313--342.
    Abstract
    We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.

  • D. Knees, R. Rossi, C. Zanini, A vanishing viscosity approach in damage mechanics, in: Variational Methods for Evolution, Workshop, December 4--10, 2011, A. Mielke, F. Otto, G. Savaré, U. Stefanelli, eds., 8 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2011, pp. 3153--3155.

  • M. Thomas, Modeling and analysis of rate-independent damage and delamination processes, in: Proceedings of the 19th International Conference on Computer Methods in Mechanics (online only), 2011, pp. 1--6.

  • P. Krejčí, E. Rocca, J. Sprekels, Liquid-solid phase transitions in a deformable container, in: Continuous Media with Microstructure, B. Albers, ed., Springer, Berlin/Heidelberg, 2010, pp. 285-300.

  • A. Mielke, Existence theory for finite-strain crystal plasticity with gradient regularization, in: IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, K. Hackl, ed., 21 of IUTAM Bookseries, Springer, Heidelberg, 2010, pp. 171--183.

  • H. Gajewski, J.A. Griepentrog, A. Mielke, J. Beuthan, U. Zabarylo, O. Minet, Image segmentation for the investigation of scattered-light images when laser-optically diagnosing rheumatoid arthritis, in: Mathematics -- Key Technology for the Future, W. Jäger, H.-J. Krebs, eds., Springer, Heidelberg, 2008, pp. 149--161.

  • A. Mielke, Numerical approximation techniques for rate-independent inelasticity, in: Proceedings of the IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, B.D. Reddy, ed., 11 of IUTAM Bookseries, Springer, 2008, pp. 53--63.

  • A. Petrov, J.A.C. Martins, M.D.P. Monteiro Marques, Mathematical results on the stability of quasi-static paths of elastic-plastic systems with hardening, in: Topics on Mathematics for Smart Systems, B. Miara, G. Stavroulakis, V. Valente, eds., World Scientific, Singapore, 2007, pp. 167--182.

  • A. Petrov, Thermally driven phase transformation in shape-memory alloys, in: Analysis and Numerics of Rate-Independent Processes, Workshop, February 26 -- March 2, 2007, 4 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2007, pp. 605--607.

  • M. Eleuteri, Some P.D.E.s with hysteresis, in: Free Boundary Problems. Theory and Applications, I.N. Figueiredo, J.F. Rodrigues, L. Santos, eds., 154 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 2007, pp. 159-168.

  • A. Glitzky, Energy models where the equations are defined on different domains, in: GAMM Annual Meeting 2006 -- Berlin, Special Issue (Vol. 6, Issue 1) of PAMM (Proceedings of Applied Mathematics and Mechanics), Wiley-VCH Verlag, Weinheim, 2006, pp. 629--630.

  • R. Hünlich, G. Albinus, H. Gajewski, A. Glitzky, W. Röpke, J. Knopke, Modelling and simulation of power devices for high-voltage integrated circuits, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 401--412.

  • I.V. Skrypnik, H. Gajewski, On the uniqueness of solutions to nonlinear elliptic and parabolic problems (in Russian), in: Differ. Uravn. i Din. Sist., dedicated to the 80th anniversary of the Academician Evgenii Frolovich Mishchenko, Suzdal, 2000, 236 of Tr. Mat. Inst. Steklova, Moscow, Russia, 2002, pp. 318--327.

  • R. Hünlich, A. Glitzky, On energy estimates for electro-diffusion equations arising in semiconductor technology, in: Partial differential equations. Theory and numerical solution, W. Jäger, J. Nečas, O. John, K. Najzar, eds., 406 of Chapman & Hall Research Notes in Mathematics, Chapman & Hall, Boca Raton, FL, 2000, pp. 158--174.

  • H.-Chr. Kaiser, J. Rehberg, About some mathematical questions concerning the embedding of Schrödinger-Poisson systems into the drift-diffusion model of semiconductor devices, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 1328--1333.

  Preprints, Reports, Technical Reports

  • J. Fuhrmann, A. Glitzky, M. Liero, Hybrid finite-volume/finite-element schemes for $p(x)$-Laplace thermistor models, Preprint no. 2378, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2378 .
    Abstract, PDF (1063 kByte)
    We introduce an empirical PDE model for the electrothermal description of organic semiconductor devices by means of current and heat flow. The current flow equation is of p(x)-Laplace type, where the piecewise constant exponent p(x) takes the non-Ohmic behavior of the organic layers into account. Moreover, the electrical conductivity contains an Arrhenius-type temperature law. We present a hybrid finite-volume/finite-element discretization scheme for the coupled system, discuss a favorite discretization of the p(x)-Laplacian at hetero interfaces, and explain how path following methods are applied to simulate S-shaped current-voltage relations resulting from the interplay of self-heating and heat flow.

  • P. Colli, J. Sprekels, Optimal boundary control of a nonstandard Cahn--Hilliard system with dynamic boundary condition and double obstacle inclusions, Preprint no. 2370, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2370 .
    Abstract, PDF (268 kByte)
    In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P.Podio-Guidugli in Ric. Mat. 55 (2006), pp.105-118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace-Beltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 35-58, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 1-30, for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials.

  • P. Colli, G. Gilardi, J. Sprekels, Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions, Preprint no. 2369, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2369 .
    Abstract, PDF (353 kByte)
    This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.

  • M. Heida, S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Preprint no. 2366, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2366 .
    Abstract, PDF (548 kByte)
    In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.

  • A. Alphonse, Ch.M. Elliott, J. Terra, A coupled ligand-receptor bulk-surface system on a moving domain: Well posedness, regularity and convergence to equilibrium, Preprint no. 2357, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2357 .
    Abstract, PDF (536 kByte)
    We prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modelling receptor-ligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the non-moving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgi-type arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for time-dependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves.

  • H. Neidhardt, A. Stephan, V.A. Zagrebnov, Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems, Preprint no. 2356, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2356 .
    Abstract, PDF (454 kByte)
    In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(J,X), consisting of X-valued functions on the time-interval J. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(J,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.

  • S. Reichelt, Corrector estimates for a class of imperfect transmission problems, Preprint no. 2354, WIAS, Berlin, 2016.
    Abstract, PDF (480 kByte)
    Based on previous homogenization results for imperfect transmission problems in two-component domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order ερ, where ε > 0 describes the periodicity of the microstructure and ρ ∈ (0 , 1/2] depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method.

  • M. Thomas, Ch. Zanini, Cohesive zone-type delamination in visco-elasticity, Preprint no. 2350, WIAS, Berlin, 2016.
    Abstract, PDF (2223 kByte)
    We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [Ortiz&Pandoli99Int.J.Numer.Meth.Eng.], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.

    Due to the presence of multivalued and unbounded operators featuring non-penetration and the `memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [Roubicek09M2AS] and refined in [Rossi&Thomas15WIAS-Preprint2113].

  • A. Mielke, M. Mittnenzweig, Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities, Preprint no. 2349, WIAS, Berlin, 2016.
    Abstract, PDF (618 kByte)
    We discuss how the recently developed energy-dissipation methods for reactiondi usion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

  • S. Dipierro, J. Serra, E. Valdinoci, Improvement of flatness for nonlocal phase transitions, Preprint no. 2345, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2345 .
    Abstract, PDF (580 kByte)
    We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations, we obtain a result in the same spirit of a celebrated theorem of Savin for the classical case. The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotropic, integro-differential operators.

  • B. Abdellaoui, A. Dieb, E. Valdinoci, A nonlocal concave-convex problem with nonlocal mixed boundary data, Preprint no. 2344, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2344 .
    Abstract, PDF (293 kByte)
    The aim of this paper is to study a nonlocal equation with mixed Neumann and Dirichlet external conditions. We prove existence, nonexistence and multiplicity of positive energy solutions and analyze the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

  • D. Belomestny, J.G.M. Schoenmakers, Projected particle methods for solving McKean--Vlaslov equations, Preprint no. 2341, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2341 .
    Abstract, PDF (320 kByte)
    We study a novel projection-based particle method to the solution of the corresponding McKean-Vlasov equation. Our approach is based on the projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method can profit from additional smoothness of the underlying density and leads in many situation to a significant reduction of numerical complexity compared to kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The case of linearly growing coefficients of the McKean-Vlasov equation turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift.

  • M. Heida, A. Mielke, Averaging of time-periodic dissipation potentials in rate-independent processes, Preprint no. 2336, WIAS, Berlin, 2016.
    Abstract, PDF (4852 kByte)
    We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε→0 and show that the effctive dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit ε→0.

  • S. Dipierro, J. Serra, E. Valdinoci, Nonlocal phase transitions: Rigidity results and anisotropic geometry, Preprint no. 2334, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2334 .
    Abstract, PDF (237 kByte)
    We provide a series of rigidity results for a nonlocal phase transition equation.

    The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi.

    Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in a companion paper.
  • A. Mielke, Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics, Preprint no. 2326, WIAS, Berlin, 2016.
    Abstract, PDF (444 kByte)
    We consider reaction-diffusion systems on a bounded domain with no-flux boundary conditions. All reactions are given by the mass-action law and are assumed to satisfy the complex-balance condition. In the case of a diagonal diffusion matrix, the relative entropy is a Liapunov functional. We give an elementary proof for the Liapunov property as well a few explicit examples for the condition of complex or detailed balancing.
     
    We discuss three methods to obtain energy-dissipation estimates, which guarantee exponential decay of the relative entropy, all of which rely on the log-Sobolev estimate and suitable handling of the reaction terms as well as the mass-conservation relations. The three methods are (i) a convexification argument based on the author's joint work with Haskovec and Markowich, (ii) a series of analytical estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness argument of developed by Glitzky and Hünlich.

  • S. Dipierro, A. Karakhanyan, E. Valdinoci, A nonlinear free boundary problem with a self-driven Bernoulli condition, Preprint no. 2325, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2325 .
    Abstract, PDF (455 kByte)
    We study a Bernoulli type free boundary problem with two phases and a nonlinear energy superposition. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases. In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem).

    Another property of this type of problems is that the minimizer in a given domain is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem.

    Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman.

    It also interpolates the classical case, in the sense that the blow-up limits are minimizers of the Alt-Caffarelli-Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit.

    We also develop a detailed optimal regularity theory for the minimizers and for their free boundaries.
  • P.-É. Druet, Analysis of improved Nernst--Planck--Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix, Preprint no. 2321, WIAS, Berlin, 2016.
    Abstract, PDF (387 kByte)
    We continue our investigations of the improved Nernst-Planck-Poisson model introduced by Dreyer, Guhlke and Müller 2013. In the paper by Dreyer, Druet, Gajewski and Guhlke 2016, the analysis relies on the hypothesis that the mobility matrix has maximal rank under the constraint of mass conservation (rank N-1 for the mixture of N species). In this paper we allow for the case that the positive eigenvalues of the mobility matrix tend to zero along with the partial mass densities of certain species. In this approach the mobility matrix has a variable rank between zero and N-1 according to the number of locally available species. We set up a concept of weak solution able to deal with this scenario, showing in particular how to extend the fundamental notion of emphdifferences of chemical potentials that supports the modelling and the analysis in Dreyer, Druet, Gajewski and Guhlke 2016. We prove the global-in-time existence in this solution class.

  • S. Dipierro, O. Savin, E. Valdinoci, Definition of fractional Laplacian for functions with polynomial growth, Preprint no. 2318, WIAS, Berlin, 2016.
    Abstract, PDF (368 kByte)
    We introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition which can be useful for applications in various fields, such as blowup and free boundary problems. In this setting, when the solution has a polynomial growth at infinity, the right hand side of the equation is not just a function, but an equivalence class of functions modulo polynomials of a fixed order. We also give a sharp version of the Schauder estimates in this framework, in which the full smooth Hölder norm of the solution is controlled in terms of the seminorm of the nonlinearity. Though the method presented is very general and potentially works for general nonlocal operators, for clarity and concreteness we focus here on the case of the fractional Laplacian.

  • S. Dipierro, F. Maggi, E. Valdinoci, Asymptotic expansions of the contact angle in nonlocal capillarity problems, Preprint no. 2315, WIAS, Berlin, 2016.
    Abstract, PDF (294 kByte)
    We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting a family of fractional interaction kernels The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient is negative, and larger if it is positive. In addition, we address the asymptotics of the fractional Young's law in the limit case s close to 0 of interaction kernels with heavy tails. Interestingly, forsmall s, the dependence of the contact angle from the relative adhesion coefficient becomes linear.

  • K. Disser, Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction, Preprint no. 2313, WIAS, Berlin, 2016.
    Abstract, PDF (302 kByte)
    We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and "entropic'' diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L-bounds and a Schaefer's fixed point argument. This allows us to extend the setting e.g. to Allen-Cahn dissipative dynamics and to include large classes of inhomogeneous boundary conditions and external forces.

  • A. Muntean, S. Reichelt, Corrector estimates for a thermo-diffusion model with weak thermal coupling, Preprint no. 2310, WIAS, Berlin, 2016.
    Abstract, PDF (1060 kByte)
    The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology ``weak thermal coupling'' refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conduction-diffusion interaction terms, while the ``high-contrast'' is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with ε-independent estimates for the thermal and concentration fields and for their coupled fluxes

  • S. Dipierro, O. Savin, E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations, Preprint no. 2303, WIAS, Berlin, 2016.
    Abstract, PDF (236 kByte)
    We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally s-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.

  • S. Patrizi, E. Valdinoci, Long-time behavior for crystal dislocation dynamics, Preprint no. 2302, WIAS, Berlin, 2016.
    Abstract, PDF (409 kByte)
    We describe the asymptotic states for the solutions of a nonlocal equation of evolutionary type, which have the physical meaning of the atom dislocation function in a periodic crystal.

    More precisely, we can describe accurately the ``smoothing effect'' on the dislocation function occurring slightly after a ``particle collision'' (roughly speaking, two opposite transitions layers average out) and, in this way, we can trap the atom dislocation function between a superposition of transition layers which, as time flows, approaches either a constant function or a single heteroclinic (depending on the algebraic properties of the orientations of the initial transition layers). <\p> The results are endowed of explicit and quantitative estimates and, as a byproduct, we show that the ODE systems of particles that governs the evolution of the transition layers does not admit stationary solutions (i.e., roughly speaking, transition layers always move).
  • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible reacting electrolytes, Preprint no. 2291, WIAS, Berlin, 2016.
    Abstract, PDF (638 kByte)

    We consider an improved Nernst-Planck-Poisson model for compressible electrolytes first proposed by Dreyer et al. in 2013. The model takes into account the elastic deformation of the medium. In particular, large pressure contributions near electrochemical interfaces induce an inherent coupling of mass and momentum transport. The model consists of convection-diffusion-reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity and the Poisson equation for the electrical potential. Cross-diffusion phenomena occur due to the principle of mass conservation. Moreover, the diffusion matrix (mobility matrix) has a zero eigenvalue, meaning that the system is degenerate parabolic. In this paper we establish the existence of a global-in- time weak solution for the full model, allowing for cross-diffusion and an arbitrary number of chemical reactions in the bulk and on the active boundary.
  • P. Colli, G. Gilardi, J. Sprekels, Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition, Preprint no. 2284, WIAS, Berlin, 2016.
    Abstract, PDF (331 kByte)
    In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.

  • E. Cinti, C. Sinestrari, E. Valdinoci, Neckpinch singularities in fractional mean curvature flows, Preprint no. 2282, WIAS, Berlin, 2016.
    Abstract, PDF (268 kByte)
    In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that, for any dimension n ≥ 2, there exist embedded hypersurfaces in Rn which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n ≥ 3. Interestingly, when n=2, our result provides instead a counterexample in the nonlocal framework to the well known Grayson's Theorem [17], which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.

  • S. Dipierro, E. Valdinoci, Nonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness, Preprint no. 2281, WIAS, Berlin, 2016.
    Abstract, PDF (456 kByte)
    We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible, referring to the papers [CRS10, SV13, CV13, BFV14, FV, DSV15, CSV16] for full details.

  • F. Maggi, E. Valdinoci, Capillarity problems with nonlocal surface tension energies, Preprint no. 2274, WIAS, Berlin, 2016.
    Abstract, PDF (377 kByte)
    We explore the possibility of modifying the classical Gauss free energy functional used in capillarity theory by considering surface tension energies of nonlocal type. The corresponding variational principles lead to new equilibrium conditions which are compared to the mean curvature equation and Young's law found in classical capillarity theory. As a special case of this family of problems we recover a nonlocal relative isoperimetric problem of geometric interest.

  • M.H. Farshbaf Shaker, Ch. Heinemann, Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D, Preprint no. 2269, WIAS, Berlin, 2016.
    Abstract, PDF (247 kByte)
    Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variable. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [M. H. Farshbaf-Shaker, C. Heinemann: A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015), 2749--2793], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, well-posedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system.

  • M. Cozzi, S. Dipierro, E. Valdinoci, Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces, Preprint no. 2264, WIAS, Berlin, 2016.
    Abstract, PDF (505 kByte)
    This paper contains three types of results: the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other).
    In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

  • M. Cozzi, S. Dipierro, E. Valdinoci, Nonlocal phase transitions in homogeneous and periodic media, Preprint no. 2262, WIAS, Berlin, 2016.
    Abstract, PDF (235 kByte)
    We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as Gamma-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media.
    Finally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.

  • R. Rossi, M. Thomas, From adhesive to brittle delamination in visco-elastodynamics, Preprint no. 2259, WIAS, Berlin, 2016.
    Abstract, PDF (735 kByte)
    In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.

  • M. Heida, Stochastic homogenization of rate-independent systems, Preprint no. 2258, WIAS, Berlin, 2016.
    Abstract, PDF (612 kByte)
    We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic two-scale convergence, which are related to classical Gamma-convergence results. The main result is a general liminf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure.

  • S. Dipierro, N. Soave, E. Valdinoci, On fractional elliptic equations in Lipschitz sets and epigraphs: Regularity, monotonicity and rigidity results, Preprint no. 2256, WIAS, Berlin, 2016.
    Abstract, PDF (534 kByte)
    We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.

    These results can be seen as the nonlocal counterpart of the celebrated article [4].

  • S. Dipierro, M. Novaga, E. Valdinoci, Rigidity of critical points for a nonlocal Ohta--Kawasaki energy, Preprint no. 2252, WIAS, Berlin, 2016.
    Abstract, PDF (153 kByte)
    We investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers.

    We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This example shows that a constraint on the diameter is, in general, necessary to establish the radial symmetry of the critical points.

  • K. Disser, A.F.M. TER Elst, J. Rehberg, On maximal parabolic regularity for non-autonomous parabolic operators, Preprint no. 2249, WIAS, Berlin, 2016.
    Abstract, PDF (335 kByte)
    We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

  • M. Bulíček, A. Glitzky, M. Liero, Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions, Preprint no. 2247, WIAS, Berlin, 2016.
    Abstract, PDF (433 kByte)
    We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

  • H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions, Preprint no. 2238, WIAS, Berlin, 2016.
    Abstract, PDF (14 MByte)
    This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

  • M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Preprint no. 2237, WIAS, Berlin, 2016.
    Abstract, PDF (10 MByte)
    A class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type.

  • J. Haskovec, S. Hittmeir, P. Markowich, A. Mielke, Decay to equilibrium for energy-reaction-diffusion systems, Preprint no. 2233, WIAS, Berlin, 2016.
    Abstract, PDF (436 kByte)
    We derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L1 using Cziszar-Kullback-Pinsker type inequalities.

  • A. Roggensack, Ch. Kraus, Existence of weak solutions for the Cahn--Hilliard reaction model including elastic effects and damage, Preprint no. 2231, WIAS, Berlin, 2016.
    Abstract, PDF (418 kByte)
    In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries. Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the coupled PDE system.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Preprint no. 2228, WIAS, Berlin, 2016.
    Abstract, PDF (294 kByte)
    In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in citeHZO. The model consists of a Cahn--Hilliard equation for the tumor cell fraction $vp$ coupled to a reaction-diffusion equation for a function $s$ representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

  • K. Disser, M. Liero, J. Zinsl, On the evolutionary Gamma-convergence of gradient systems modeling slow and fast chemical reactions, Preprint no. 2227, WIAS, Berlin, 2016.
    Abstract, PDF (489 kByte)
    We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Γ-convergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudo-metric.

  • E. Cinti, J. Serra, E. Valdinoci, Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces, Preprint no. 2223, WIAS, Berlin, 2016.
    Abstract, PDF (695 kByte)
    We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $nge 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_1/2$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $R^3$. On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n=2,3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ ---with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

  • L. Caffarelli, S. Dipierro, E. Valdinoci, A logistic equation with nonlocal interactions, Preprint no. 2216, WIAS, Berlin, 2016.
    Abstract, PDF (329 kByte)
    We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: beginitemize item bounded domains, item periodic environments, item transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. enditemize In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

  • A. Farina, E. Valdinoci, Anisotropic nonlocal operators regularity and rigidity theorems for a class of anisotropic nonlocal operators, Preprint no. 2213, WIAS, Berlin, 2016.
    Abstract, PDF (284 kByte)
    We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order $2$ in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.

  • S. Dipierro, A. Karakhanyan, E. Valdinoci, A class of unstable free boundary problems, Preprint no. 2212, WIAS, Berlin, 2016.
    Abstract, PDF (395 kByte)
    We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy. In sharp contrast with the linear case, the problem considered in this paper is unstable, namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain. We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution. As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problems. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possibly nonlocality of the problem, but it is due to the nonlinear character of the energy functional.

  • B. Albers, P. Krejčí, E. Rocca, Solvability of an unsaturated porous media flow problem with thermomechanical interaction, Preprint no. 2211, WIAS, Berlin, 2016.
    Abstract, PDF (259 kByte)
    A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.

  • CH. Heinemann, K. Sturm, Shape optimisation for a class of semilinear variational inequalities with applications to damage models, Preprint no. 2209, WIAS, Berlin, 2016.
    Abstract, PDF (588 kByte)
    The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields.

  • M. Sáez, E. Valdinoci, On the evolution by fractional mean curvature, Preprint no. 2183, WIAS, Berlin, 2015.
    Abstract, PDF (332 kByte)
    In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain quantities (such as positive fractional curvature) and smoothness of graphical evolutions.

  • N. Ahmed, T.Ch. Rebollo, V. John, S. Rubino, Analysis of a full space-time discretization of the Navier--Stokes equations by a local projection stabilization method, Preprint no. 2166, WIAS, Berlin, 2015, DOI 10.20347/WIAS.PREPRINT.2166 .
    Abstract, PDF (649 kByte)
    A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier--Stokes equations is presented. The focus is on the high-order term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent flows. Smooth unsteady flows are simulated with optimal order of accuracy.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, Preprint no. 2165, WIAS, Berlin, 2015, DOI 10.20347/WIAS.PREPRINT.2165 .
    Abstract, PDF (363 kByte)
    We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

  • CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Preprint no. 2164, WIAS, Berlin, 2015.
    Abstract, Postscript (1675 kByte), PDF (742 kByte)
    In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179--211]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007), 461--490] in the framework of Fourier-Navier-Stokes systems and then recently employed in [E. Feireisl, H. Petzeltová, E. Rocca: Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009), 1345--1369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 2519--2586] for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.

  • S. Dipierro, N. Soave, E. Valdinoci, On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data, Preprint no. 2152, WIAS, Berlin, 2015.
    Abstract, PDF (355 kByte)
    We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar´e-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.

  • M. Dai, E. Feireisl, E. Rocca, G. Schimperna, M.E. Schonbek , Analysis of a diffuse interface model of multispecies tumor growth, Preprint no. 2150, WIAS, Berlin, 2015.
    Abstract, PDF (203 kByte)
    We consider a diffuse interface model for tumor growth recently proposed in citecwsl. In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn-Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity $vu$ satisfies $vucdotnu>0$, where $nu$ is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.

  • E. Bonetti, E. Rocca, Generalized gradient flow structure of internal energy driven phase field systems, Preprint no. 2144, WIAS, Berlin, 2015.
    Abstract, PDF (186 kByte)
    In this paper we introduce a general abstract formulation of a variational thermomechanical model, by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, including the thermal one. In particular, choosing as thermal variable the entropy of the system, and as driving functional the internal energy, we get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. We prove a global in time existence of (weak) solutions result for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals.

  • A. Roggensack, Well-posedness of space and time dependent transport equations on a network, Preprint no. 2138, WIAS, Berlin, 2015.
    Abstract, PDF (378 kByte)
    This article is concerned with the study of weak solutions of a linear transport equation on a bounded domain with coupled boundary data for general non smooth space and time dependent velocity fields. The existence of solutions, its uniqueness and the continuous dependence of the solution on the initial and boundary data as well on the velocity is proven. The results are based on the renormalization property. At the end, the theory is shown to be applicable to the continuity equation on a network.

  • L. Caffarelli, S. Patrizi, V. Quitalo, On a long range segregation model, Preprint no. 2137, WIAS, Berlin, 2015.
    Abstract, PDF (1185 kByte)
    Segregation phenomena occurs in many areas of mathematics and science: from equipartition problems in geometry, to social and biological processes (cells, bacteria, ants, mammals) to finance (sellers and buyers). There is a large body of literature studying segregation models where the interaction between species is punctual. There are many processes though, where the growth of a population at a point is inhibited by the populations in a full area surrounding that point. This work is a first attempt to study the properties of such a segregation process.

  • V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, E. Rocca, Sliding modes for a phase-field system, Preprint no. 2133, WIAS, Berlin, 2015.
    Abstract, PDF (295 kByte)
    In the present contribution the sliding mode control (SMC) problem for a phase-field model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the state- feedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target  $phi^*$. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.

  • R. Rossi, M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, Preprint no. 2123, WIAS, Berlin, 2015.
    Abstract, PDF (889 kByte)
    We address the analysis of an abstract system coupling a rate-independet process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.

  • S. Dipierro, M. Medina, E. Valdinoci, Fractional elliptic problems with critical growth in the whole of (R^n), Preprint no. 2118, WIAS, Berlin, 2015.
    Abstract, PDF (683 kByte)
    We study a nonlinear and nonlocal elliptic equation. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this paper, we find a local minimum and a mountain pass solution of this functional. One of the crucial ingredient is a Concentration-Compactness principle. Some difficulties arise from the nonlocal structure of the problem and from the fact that we deal with an equation in the whole of the space (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.

  • F. Hamel, E. Valdinoci, A one-dimensional symmetry result for solutions of an integral equation of convolution type, Preprint no. 2115, WIAS, Berlin, 2015.
    Abstract, PDF (196 kByte)
    We consider an integral equation in the plane, in which the leading operator is of convolution type, and we prove that monotone (or stable) solutions are necessarily one-dimensiona.l

  • M. Liero, S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, Preprint no. 2114, WIAS, Berlin, 2015.
    Abstract, PDF (455 kByte)
    In this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

  • M. Cozzi, E. Valdinoci, Plane-like minimizers for a non-local Ginzburg--Landau-type energy in a periodic medium, Preprint no. 2108, WIAS, Berlin, 2015.
    Abstract, PDF (1768 kByte)
    We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.

  • F. Punzo, E. Valdinoci, Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients, Preprint no. 2104, WIAS, Berlin, 2015.
    Abstract, PDF (274 kByte)
    We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.

  • L. Rossi, A. Tellini, E. Valdinoci, The effect on Fisher--KPP propagation in a cylinder with fast diffusion on the boundary, Preprint no. 2103, WIAS, Berlin, 2015.
    Abstract, PDF (826 kByte)
    In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $R^N+1$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them.

  • S. Dipierro, E. Valdinoci, Continuity and density results for a one-phase nonlocal free boundary problem, Preprint no. 2101, WIAS, Berlin, 2015.
    Abstract, PDF (358 kByte)
    We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.

  • E. Bonetti, E. Rocca, G. Schimperna, R. Scala, On the strongly damped wave equation with constraint, Preprint no. 2094, WIAS, Berlin, 2015.
    Abstract, PDF (246 kByte)
    A weak formulation for the so-called semilinear strongly damped wave equation with constraint is introduced and a corresponding notion of solution is de?ned. The main idea in this approach consists in the use of duality techniques in Sobolev-Bochner spaces, aimed at providing a suitable "relaxation" of the constraint term. A global in time existence result is proved under the natural condition that the initial data have finite "physical" energy.

  • A. Massaccesi, E. Valdinoci, Is a nonlocal diffusion strategy convenient for biological populations in competition?, Preprint no. 2092, WIAS, Berlin, 2015.
    Abstract, PDF (307 kByte)
    We study the convenience of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances - namely, a precise condition on the distribution of the resource - under which a nonlocal dispersal behavior is favored. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local population becomes linearly unstable. In a sense, this example shows that nonlocal strategies can become successful even in an environment in which purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse. Indeed, the example considered presents a high variance of the distribu- tion of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic ingredients that favor nonlocal strategies.

  • X. Ros-Oton, J. Serra, E. Valdinoci, Pohozaev identities for anisotropic integro-differential operators, Preprint no. 2080, WIAS, Berlin, 2015.
    Abstract, PDF (339 kByte)
    We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order 2s, with s ϵ (0, 1). These identities involve local boundary terms, in which the quantity u/ds ∂Ω plays the role that ∂u/∂v plays in the second order case. Here, u is any solution to Lu = f (x, u) in Ω, with u = 0 in Rn \ Ω , and d is the distance to ∂Ω.

  Talks, Poster

  • S. Reichelt, Corrector estimates for imperfect transmission problems, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

  • S. Reichelt, tba, Analysis-Seminars Augsburg-München (ASAM), Universität Augsburg, tba, May 23, 2017.

  • M. Liero, On entropy-transport problems and the Hellinger--Kantorovich distance, Seminar of Team EDP-AIRSEA-CVGI, Université Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble, France, January 26, 2017.

  • E. Valdinoci, Nonlocal Equations and Applications, Spring School on Nonlinear PDEs and Related Problems, January 15 - 19, 2016, African Institute for Mathematical Sciences (AIMS), Mbour, Senegal.

  • P.-É. Druet, Analysis of recent Nernst--Planck--Poisson--Navier--Stokes systems of electrolytes, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6 - 10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

  • P.-É. Druet, Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible electrolytes, Seminar EDE, Institute of Mathematics, Department of Evolution Differential Equations (EDE), Prague, Czech Republic, January 10, 2017.

  • TH. Koprucki, Mathematical knowledge management as a route to sustainability in mathematical modeling and simulation, 2nd Leibniz MMS Days 2017, February 22 - 24, 2017, Technische Informationsbibliothek (TIB), Hannover, February 22, 2017, DOI 10.5446/21908 .

  • A. Mielke, A geometric approach to reaction-diffusion equations, Institutskolloquium, Universität Potsdam, Institut für Mathematik, Potsdam, January 25, 2017.

  • A. Mielke, Optimal transport versus growth and decay, International Conference ``Calculus of Variations and Optimal Transportation'' in the Honor of Yann Brenier for his 60th Birthday, January 9 - 11, 2017, Institut Henri Poincaré, Paris, France, January 11, 2017.

  • A. Mielke, Uniform exponential decay for energy-reaction-diffusion systems, Analysis Seminar, University of Pavia, Department of Mathematics, Italy, March 21, 2017.

  • J. Rehberg, On optimal elliptic Sobolev regularity, Oberseminar Prof. Ira Neitzel, Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Numerische Simulation, Bonn, February 2, 2017.

  • E. Cinti, Quantitative flatness results and BV-estimates for nonlocal minimal surfaces, Seminario di Equazioni Differenziali, Università degli Studi di Roma ``Tor Vergata'', Italy, April 19, 2016.

  • K. Disser, Convergence for gradient systems of slow and fast chemical reactions, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 11, 2016.

  • K. Disser, E-convergence to the quasi-steady-state approximation in systems of chemical reactions, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 25, 2016.

  • S. Reichelt, Error estimates for elliptic and parabolic equations with oscillating coefficients, Karlstad Applied Analysis Seminar, Karlstad University, Department of Mathematics and Computer Science, Sweden, April 13, 2016.

  • S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary $Gamma$-convergence, Joint Annual Meeting of DMV and GAMM, Young Researchers' Minisymposium ``Multiscale Evolutionary Problems'', March 7 - 11, 2016, Technische Universität Braunschweig, March 7, 2016.

  • S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, Workshop ``Patterns of Dynamics'', Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 25 - 29, 2016.

  • S. Reichelt, Homogenization of Cahn--Hilliard-type equations via gradient structures, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 2 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equation'', July 1 - 5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 3, 2016.

  • TH. Frenzel, EDP-convergence for delamination and a wiggly energy model, 2nd Berlin Dresden Prague Würzburg Workshop on Mathematics of Continuum Mechanics, Technische Universität Dresden, Fachbereich Mathematik, Dresden, December 5, 2016.

  • TH. Frenzel, Evolutionary Gamma-convergence for a delamination model, Workshop on Industrial and Applied Mathematics 2016, 5th Symposium of German SIAM Student Chapters, August 31 - September 2, 2016, University of Hamburg, Department of Mathematics, Hamburg, September 1, 2016.

  • TH. Frenzel, Evolutionary Gamma-convergence for amplitude equations and for wiggly energy models, Winter School 2016: Calculus of Variations in Physics and Materials Science, Würzburg, February 15 - 19, 2016.

  • TH. Frenzel, Examples of evolutionary Gamma-convergence, Workshop on Industrial and Applied Mathematics 2016, 5th Symposium of German SIAM Student Chapters, Hamburg, August 31 - September 2, 2016.

  • M. Heida, Large deviation principle for a stochastic Allen--Cahn equation, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 25, 2016.

  • M. Liero, Gradient structures for reaction-diffusion systems and optimal entropy-transport problems, Workshop ``Variational and Hamiltonian Structures: Models and Methods'', July 11 - 15, 2016, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, July 11, 2016.

  • M. Liero, OLEDs - a hot matter? Electrothermal modeling of OLEDs., sc Matheon Workshop, 9th Annual Meeting ``Photonic Devices'', March 3 - 4, 2016, Zuse Institute Berlin, Berlin, March 4, 2016.

  • M. Liero, On $p(x)$-Laplace thermistor models describing eletrothermal feedback in organic semiconductors, The 19th European Conference on Mathematics for Industry (ECMI 2016), Minisymposium 23 ``Charge Transport in Semiconductor Materials: Emerging and Established Mathematical Topics'', June 13 - 17, 2016, Universidade de Santiago de Compostela, Spain, June 15, 2016.

  • M. Liero, On $p(x)$-Laplace thermistor models describing eletrothermal feedback in organic semiconductors, Joint Annual Meeting of DMV and GAMM, Section 14 ``Applied Analysis'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 9, 2016.

  • M. Liero, On electrothermal feedback in organic light-emitting diodes, Berlin Dresden Prague Würzburg Workshop ``Mathematics of Continuum Mechanics'', Technische Universität Dresden, Fachbereich Mathematik, December 5, 2016.

  • M. Liero, On geodesic curves and convexity of functionals with respect to the Hellinger--Kantorovich distance, Workshop ``Optimal Transport and Applications'', November 7 - 11, 2016, Scuola Normale Superiore, Dipartimento di Matematica, Pisa, Italy, November 10, 2016.

  • E. Valdinoci, A notion of fractional perimeter and nonlocal minimal surfaces, Seminar, Universitá del Salento, Dipartimento di Matematics e Fisica ``Ennio de Giorgi'', Lecce, Italy, June 22, 2016.

  • E. Valdinoci, Interior and boundary properties of nonlocal minimal surfaces, Calcul des Variations & EDP, Université Aix-Marseille, Institut de Mathématiques de Marseille, France, February 25, 2016.

  • E. Valdinoci, Interior and boundary properties on nonlocal minimal surfaces, 3rd Conference on Nonlocal Operators and Partial Differential Equations, June 27 - July 1, 2016, Bedlewo, Poland, June 27, 2016.

  • E. Valdinoci, Nonlocal equations and applications, Calculus of Variations and Nonlinear Partial Differential Equations, May 16 - 27, 2016, Columbia University, Department of Mathematics, New York, USA, May 25, 2016.

  • E. Valdinoci, Nonlocal equations from different points of view, Research Seminar, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, August 31, 2016.

  • E. Valdinoci, Nonlocal equations from various perspectives, PIMS Workshop on Nonlocal Variational Problems and PDEs, June 13 - 17, 2016, University of British Columbia, Vancouver, Canada, June 13, 2016.

  • E. Valdinoci, Nonlocal minimal surfaces, Analysis, PDEs, and Geometry Seminar, Monash University, Clayton, Australia, August 9, 2016.

  • E. Valdinoci, Nonlocal minimal surfaces and phase transitions, Seminar, University of Leeds, School of Mathematics, UK, April 19, 2016.

  • E. Valdinoci, Nonlocal minimal surfaces, a geometric and analytic insight, Seminar on Differential Geometry and Analysis, Otto-von-Guericke-Universität Magdeburg, January 18, 2016.

  • E. Valdinoci, Nonlocal minimal surfaces: Regularity and quantitative properties, Conference on Recent Trends on Elliptic Nonlocal Equations, The Fields Institute for Research in Mathematical Sciences, Toronto, Canada, June 9, 2016.

  • E. Valdinoci, Planelike solutions in periodic media, Asymptotic Patterns in Variational Problems: PDE and Geometric Aspects, September 25 - 30, 2016, Banff International Research Station for Mathematical Innovation and Discovery, Oaxaca, Mexico, September 28, 2016.

  • E. Valdinoci, Superfici minime e transizioni di fase (non)locali, EDP e Dintorni II Incontro, Università di Bari, Dipartimento di Matematica, Italy, May 13, 2016.

  • M. Becker, Th. Frenzel, Th. Niedermayer, S. Reichelt, M. Bär, A. Mielke, Competing patterns in anti-symmetrically coupled Swift--Hohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

  • A. Glitzky, $p(x)$-Laplace thermistor models for electrothermal effects in organic semiconductor devices, 7th European Congress of Mathematics (7ECM), Minisymposium 22 ``Mathematical Methods for Semiconductors'', July 18 - 22, 2016, Technische Universität Berlin, July 22, 2016.

  • A. Glitzky, $p(x)$-Laplace thermistor models for electrothermal feedback in organic semiconductor devices, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 2016.

  • M. Thomas, Analysis and optimization for edge-emitting semiconductor heterostructures, 7th European Congress of Mathematics (ECM), session CS-8-A, July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 19, 2016.

  • M. Thomas, Analysis and optimization for edge-emitting semiconductor heterostructures, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 2 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equation'', July 1 - 5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 3, 2016.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Delamination models in visco-elastodynamics, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, June 10, 2016.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, 7th European Congress of Mathematics (ECM), minisymposium ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 22, 2016.

  • M. Thomas, Energetic concepts for coupled rate-independent and rate-dependent processes: Damage & delamination in visco-elastodynamics, International Conference ``Mathematical Analysis of Continuum Mechanics and Industrial Applications II'' (CoMFoS16), October 22 - 24, 2016, Kyushu University, Fukuoka, Japan.

  • M. Thomas, From adhesive contact to brittle delamination in visco-elastodynamics, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, special session ``Rate-dependent and Rate-independent Evolution Problems in Continuum Mechanics: Analytical and Numerical Aspects'', July 1 - 5, 2016, The American Institute of Mathematical Sciences, Orlando (Florida), USA, July 4, 2016.

  • M. Thomas, From adhesive contact to brittle delamination in visco-elastodynamics, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 26, 2016.

  • M. Thomas, Non-smooth PDEs in material failure: Towards dynamic fracture, Joint Annual Meeting of DMV and GAMM, Section 14 ``Applied Analysis'', March 7 - 11, 2016, Technische Universität Braunschweig, March 10, 2016.

  • M. Thomas, Rate-independent evolution of sets, INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5 - 8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 6, 2016.

  • M. Thomas, Rate-independent evolution of sets & application to fracture processes, Seminar on Analysis, Kanazawa University, Institute of Science and Engineering, Kanazawa, Japan, October 28, 2016.

  • N. Ahmed, On the grad-div stabilization for the steady Oseen and Navier--Stokes evaluations, International Conference of Boundary and Interior Layers (BAIL 2016), August 15 - 19, 2016, Beijing Computational Science Research Center, Beijing, China, August 15, 2016.

  • P.-É. Druet, Existence of global weak solutions for generalized Poisson--Nernst--Planck systems, 7th European Congress of Mathematics (ECM), minisymposium ``Analysis of Thermodynamically Consistent Models of Electrolytes in the Context of Battery Research'', July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

  • S.P. Frigeri, On a diffuse interface model of tumor growth, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 2016.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, Salzburg Mathematics Colloquium, Universität Salzburg, Fachbereich Mathematik, Austria, June 9, 2016.

  • M. Hintermüller, Recent trends in optimal control problems with nonsmooth structures, Computational Methods for Control of Infinite-dimensional Systems, March 14 - 18, 2016, Institute for Mathematics and its Applications, Minneapolis, USA, March 14, 2016.

  • A. Mielke, Entropy-entropy production estimates for energy-reaction diffusion systems, Workshop ``Forefront of PDEs: Modelling, Analysis and Numerics'', December 12 - 14, 2016, Technische Universität Wien, Institut für Analysis and Scientific Computing, Austria, December 13, 2016.

  • A. Mielke, Evolution driven by energy and entropy, SFB1114 Kolloquium, Freie Universität Berlin, Berlin, June 30, 2016.

  • A. Mielke, Evolutionary Gamma-convergence, 2nd CENTRAL School on Analysis and Numerics for Partial Differential Equations, August 29 - September 2, 2016, Humboldt-Universität zu Berlin, Institut für Mathematik.

  • A. Mielke, Exponential decay into thermodynamical equilibrium for reaction-diffusion systems with detailed balance, Workshop ``Patterns of Dynamics'', July 25 - 29, 2016, Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 28, 2016.

  • A. Mielke, Global existence for finite-strain viscoplasticity via the energy-dissipation principle, Seminar ``Analysis & Mathematical Physics'', Institute of Science and Technology Austria (IST Austria), Vienna, Austria, July 7, 2016.

  • A. Mielke, Gradient structures and dissipation distances for reaction-diffusion equation, Mathematisches Kolloquium, Westfälische Wilhelms-Universität, Institut für Mathematik, Münster, April 28, 2016.

  • A. Mielke, Mutual recovery sequences and evolutionary relaxation of a two-phase problem, 2nd Workshop on CENTRAL Trends in Analysis and Numerics for PDEs, May 26 - 28, 2016, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic, May 27, 2016.

  • A. Mielke, On the Hellinger--Kantorovich distance for reaction and diffusion, Workshop ``Interactions between Partial Differential Equations & Functional Inequalities'', September 12 - 16, 2016, The Royal Swedish Academy of Sciences, Institut Mittag--Leffler, Stockholm, Sweden, September 12, 2016.

  • A. Mielke, On the geometry of reaction and diffusion, INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, September 5 - 8, 2016, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Rome, Italy, September 7, 2016.

  • A. Mielke, Optimal transport versus reaction --- On the geometry of reaction-diffusion equations, Pure Analysis and PDEs Seminar, Imperial College London, Department of Mathematics, UK, May 12, 2016.

  • J. Rehberg, On Hölder continuity for elliptic and parabolic problems, 8th Singular Days, June 27 - 30, 2016, University of Lorraine, Department of Sciences and Technologies, Nancy, France, June 29, 2016.

  • J. Rehberg, On non-smooth parabolic equations, Oberseminar Analysis, Universität Kassel, Institut für Mathematik, May 2, 2016.

  • J. Rehberg, On non-smooth parabolic equations, Oberseminar Analysis, Leibniz Universität Hannover, Institut für Angewandte Mathematik, May 10, 2016.

  • J. Rehberg, On optimal elliptic Sobolev regularity, Oberseminar Analysis, Technische Universität Clausthal, Institut für Mathematik, Clausthal-Zellerfeld, January 20, 2016.

  • E. Valdinoci, Nonlocal equations in phase transitions, minimal surfaces, crystallography and life sciences, Escuela CAPDE Ecuaciones diferenciales parciales no lineales, Universidad de Chile, Facultad de Ciencias Físicas y Matemáticas, Santiago de Chile, Chile.

  • E. Cinti, A quantitative weighted isoperimetric inequality via the ABP method, Oberseminar Analysis, Universität Bonn, Institut für Angewandte Mathematik, February 5, 2015.

  • E. Cinti, Quantitative flatness results for nonlocal minimal surfaces in low dimensions, Theory of Applications of Partial Differential Equations (PDE 2015), November 30 - December 4, 2015, WIAS Berlin, Berlin, December 2, 2015.

  • E. Cinti, Quantitative isoperimetric inequality via the ABP method, Università di Bologna, Dipartimento di Matematica, Bologna, Italy, July 17, 2015.

  • K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Mathematical Thermodynamics of Complex Fluids, June 29 - July 3, 2015, Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 30, 2015.

  • K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Workshop ``Young Researchers in Fluid Dynamics'', June 18 - 19, 2015, Technische Universität Darmstadt, Fachbereich Mathematik, Darmstadt, June 18, 2015.

  • K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Seminar ``Dynamische Systeme'', Technische Universität München, Zentrum Mathematik, München, February 2, 2015.

  • K. Disser, Asymptotic behavior of a rigid body with a cavity filled by a viscous liquid, Oberseminar ``Analysis'', Universität Kassel, Institut für Mathematik, Kassel, January 12, 2015.

  • K. Disser, Dynamik von Starrkörpern, die mit einer Flüssigkeit gefüllt sind, und das Ei-Problem, Mathematisches Kolloquium, Heinrich-Heine Universität Düsseldorf, Institut für Mathematik, Düsseldorf, April 10, 2015.

  • S. Patrizi, Dislocations dynamics: From microscopic models to macroscopic crystal plasticity, Analysis Seminar, The University of Texas at Austin, Department of Mathematics, USA, January 21, 2015.

  • S. Patrizi, Dislocations dynamics: From microscopic models to macroscopic crystal plasticity, Seminar, King Abdullah University of Science and Technologie, SRI -- Center for Uncertainty Quantification in Computational Science & Engineering, Jeddah, Saudi Arabia, March 25, 2015.

  • S. Patrizi, On a long range segregation model, Seminar, Università degli Studi di Salerno, Dipartimento di Matematica, Italy, May 19, 2015.

  • S. Patrizi, On a long range segregation model, Seminario di Analisi Matematica, Sapienza Università di Roma, Dipartimento di Matematica ``Guido Castelnuovo'', Italy, April 20, 2015.

  • E. Rocca, Optimal control of a nonlocal convective Cahn--Hilliard equation by the velocity, Numerical Analysis Seminars, Durham University, UK, March 13, 2015.

  • S.P. Frigeri, On a diffuse interface model of tumor growth, INdAM Workshop ``Special Materials in Complex Systems -- SMaCS 2015'', May 18 - 22, 2015, Rome, Italy, May 22, 2015.

  • S.P. Frigeri, On a nonlocal diffuse interface model for binary incompressible fluids with different densities, Mathematical Thermodynamics of Complex Fluids, June 28 - July 3, 2015, Fondazione CIME ``Roberto Conti'' (International Mathematical Summer Center), Cetraro, Italy, July 2, 2015.

  • S.P. Frigeri, Recent results on optimal control for Cahn--Hilliard/Navier--Stokes systems with nonlocal interactions, Control Theory and Related Topics, April 13 - 14, 2015, Politecnico di Milano, Italy, April 13, 2015.

  • M. Heida, Stochastic homogenization of Prandtl--Reuss plasticity, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30 - October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 1, 2015.

  • CH. Heinemann, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, 3rd Workshop of the GAMM Activity Group Analysis of Partial differential Equations, September 30 - October 2, 2015, Universität Kassel, Institut für Mathematik, October 1, 2015.

  • CH. Heinemann, On elastic Cahn--Hilliard systems coupled with evolution inclusions for damage processes, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Young Researchers' Minisymposium 2, March 23 - 27, 2015, Lecce, Italy, March 23, 2015.

  • CH. Heinemann, Solvability of differential inclusions describing damage processes and applications to optimal control problems, Universität Essen-Duisburg, Fakultät für Mathematik, Essen, December 3, 2015.

  • M. Landstorfer, Theory, structure and experimental justification of the metal/electrolyte interface, Minisymposium `` Recent Developments on Electrochemical Interface Modeling'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10 - 14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 11, 2015.

  • M. Liero, Electrothermal modeling of large-area OLEDs, sc Matheon Center Days, April 20 - 21, 2015, Technische Universität Berlin, Institut für Mathematik, Berlin, April 20, 2015.

  • M. Liero, On $p(x)$-Laplace thermistor models describing electrothermal feedback in organic semiconductor devices, Theory of Applications of Partial Differential Equations (PDE 2015), November 30 - December 4, 2015, WIAS Berlin, Berlin, December 3, 2015.

  • M. Liero, On a PDE thermistor system for large-area OLEDs, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2015), March 11 - 13, 2015, WIAS Berlin, Berlin, March 12, 2015.

  • M. Liero, On dissipation distances for reaction-diffusion equations --- The Hellinger--Kantorovich distance, Workshop ``Collective Dynamics in Gradient Flows and Entropy Driven Structures'', June 1 - 5, 2015, Gran Sasso Science Institute, L'Aquila, Italy, June 3, 2015.

  • D. Peschka, Droplets on liquids and their long way into equilibrium, Minisymposium ``Recent Progress in Modeling and Simulation of Multiphase Thin-film Type Problems'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10 - 14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 12, 2015.

  • D. Peschka, Thin-film equations with free boundaries, Jahrestagung der Deutschen Mathematiker-Vereinigung, Minisymposium ``Mathematics of Fluid Interfaces'', September 21 - 25, 2015, Universität Hamburg, Fakultät für Mathematik, Informatik und Naturwissenschaften, Hamburg, September 23, 2015.

  • D. Puzyrev, Multistability and bifurcations of laser cavity solitons induced by delayed feedback, International Workshop ``Nonlinear Photonics: Theory, Materials, Applications'', Session ``Laser Dynamics'', June 29 - July 2, 2015, Saint-Petersburg State University, Departmetnt of General Physics, Russian Federation, July 1, 2015.

  • E. Valdinoci, Dislocation dynamics in crystals: Nonlocal effects, collisions and relaxation, Mostly Maximum Principle, September 16 - 18, 2015, Castello Aragonese, Agropoli, Italy, September 16, 2015.

  • E. Valdinoci, Dislocation dynamics in crystals: Nonlocal effects, collisions and relaxation, Second Workshop on Trends in Nonlinear Analysis, September 24 - 26, 2015, GNAMPA, Universitá degli Studi die Cagliari, Dipartimento di Matematica e Informatica, Cagliari, Italy, September 26, 2015.

  • E. Valdinoci, Minimal surfaces and phase transitions with nonlocal interactions, Analysis Seminar, University of Edinburgh, School of Mathematics, UK, March 23, 2015.

  • E. Valdinoci, Nonlocal Problems in Analysis and Geometry, 2° Corso Intensivo di Calcolo delle Variazioni, June 15 - 20, 2015, Dipartimento di Matematica e Informatica di Catania, Italy.

  • E. Valdinoci, Nonlocal minimal surfaces, Seminario di Calcolo delle Variazioni & Equazioni alle Derivate Parziali, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica ``Ulisse Dini'', Italy, March 13, 2015.

  • E. Valdinoci, Nonlocal problems -- Theory and applications, School/Workshop ``Phase Transition Problems and Nonlinear PDEs'', March 9 - 11, 2015, Università di Bologna, Dipartimento di Matematica.

  • E. Valdinoci, Nonlocal problems and applications, Summer School on ``Geometric Methods for PDEs and Dynamical Systems'', June 8 - 11, 2015, École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées and Institut de Mathématiques, Equipe d'Analyse, Université Bordeaux 1, Porquerolles, France.

  • E. Valdinoci, Some models arising in crystal dislocations, Global Dynamics in Hamiltonian Systems, June 28 - July 4, 2015, Universitat Politècnica de Catalunya (BarcelonaTech), Girona, Spain, June 29, 2015.

  • E. Valdinoci, What is the (fractional) Laplacian?, Perlen-Kolloquium, Universität Basel, Fachbereich Mathematik, Switzerland, May 22, 2015.

  • S. Yanchuk, Multiscale jittering in oscillators with pulsatile delayed feedback, Short Thematic Program on Delay Differential Equations, May 11 - 15, 2015, The Fields Institute for Research in Mathematical Sciences, Toronto, Canada, May 14, 2015.

  • S. Yanchuk, Pattern formation in systems with multiple delayed feedbacks, Minisymposium ``Time-delayed Feedback'' of the SIAM Conference on Applications of Dynamical Systems, May 17 - 21, 2015, Society for Industrial and Applied Mathematics, Snowbird/Utah, USA, May 20, 2015.

  • S. Yanchuk, Stability of plane wave solutions in complex Ginzburg--Landau equation with delayed feedback, Jahrestagung der Deutschen Mathematiker-Vereinigung, Minisymposium ``Topics in Delay Differential Equations'', September 21 - 25, 2015, Universität Hamburg, Fakultät für Mathematik, Informatik und Naturwissenschaften, Hamburg, September 22, 2015.

  • S. Yanchuk, Time delays and plasticity in neuronal networks, Seminar ``Self-assembly and Self-organization in Computer Science and Biology'', September 27 - October 2, 2015, Leibniz-Zentrum für Informatik, Schloss Dagstuhl, September 28, 2015.

  • J. Sprekels, Optimal boundary control problems for Cahn--Hilliard systems with dynamic boundary conditions, INdAM Workshop ``Special Materials in Complex Systems -- SMaCS 2015'', May 18 - 22, 2015, Rome, Italy, May 21, 2015.

  • A. Glitzky, Analysis of $p(x)$-Laplace thermistor models for electrothermal feedback in organic semiconductor devices, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30 - October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, September 30, 2015.

  • A. Glitzky, Finite volume discretized reaction-diffusion systems in heterostructures, Conference on Partial Differential Equations, March 25 - 29, 2015, Technische Universität München, Zentrum Mathematik, München, March 28, 2015.

  • M. Thomas, Analysis for edge-emitting semiconductor heterostructures, Minisymposium ``Numerical and Analytical Aspects in Semiconductor Theory'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10 - 14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 10, 2015.

  • M. Thomas, Analysis of nonsmooth PDE systems with application to material failure---towards dynamic fracture, Minisymposium ``Analysis of Nonsmooth PDE Systems with Application to Material Failure'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 10 - 14, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 12, 2015.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Pavia, Italy, March 5, 2015.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Evolutionary Gamma-convergence results, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Session on Applied Analysis, March 23 - 27, 2015, Università del Salento, Lecce, Italy, March 26, 2015.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Existence and evolutionary Gamma convergence, INdAM Workshop ``Special Materials in Complex Systems -- SMaCS 2015'', May 18 - 22, 2015, Rome, Italy, May 19, 2015.

  • M. Thomas, Coupling rate-independent and rate-dependent processes: Existence results, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), GAMM Juniors Poster Session, Lecce, Italy, March 23 - 27, 2015.

  • M. Thomas, Evolutionary Gamma convergence with application to damage and delamination, Seminar DICATAM, Università di Brescia, Dipartimento di Matematica, Brescia, Italy, June 3, 2015.

  • M. Thomas, Rate-independent damage models with spatial BV-regularization --- Existence & fine properties of solutions, Oberseminar ``Angewandte Analysis'', Universität Freiburg, Abteilung für Angewandte Mathematik, Freiburg, February 10, 2015.

  • M.H. Farshbaf Shaker, A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions, National Institute for Mathematical Sciences, Division of Computational Mathematics, Daejeon, Korea (Republic of), May 20, 2015.

  • M.H. Farshbaf Shaker, Multi-material phase field approach to structural topology optimization and its relation to sharp interface approach, University of Tokyo, Graduate School of Mathematical Sciences, Japan, October 6, 2015.

  • M.H. Farshbaf Shaker, Relating phase field and sharp interface approaches to structural topology optimization, National Institute for Mathematical Sciences, Division of Computational Mathematics, Daejeon, Korea (Republic of), May 13, 2015.

  • M.H. Farshbaf Shaker, Relating phase field and sharp interface approaches to structural topology optimization, Technische Universität Berlin, Institut für Mathematik, February 5, 2015.

  • CH. Heinemann, Well-posedness of strong solutions for a damage model in 2D, Universitá di Brescia, Department DICATAM -- Section of Mathematics, Italy, March 13, 2015.

  • A. Mielke, A mathematical approach to finite-strain viscoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 20, 2015.

  • A. Mielke, Abstract approach to energetic solutions for rate-independent solutions, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 18, 2015.

  • A. Mielke, Evolutionary $Gamma$-convergence for generalized gradient systems, Workshop ``Gradient Flows'', June 22 - 23, 2015, Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, France, June 22, 2015.

  • A. Mielke, Evolutionary relaxation of a two-phase model, Mini-Workshop ``Scales in Plasticity'', November 8 - 14, 2015, Mathematisches Forschungsinstitut Oberwolfach, November 11, 2015.

  • A. Mielke, Existence results in finite-strain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 19, 2015.

  • A. Mielke, Geometric approaches at and for theoretical and applied mechanics, Phil Holmes Retirement Celebration, October 8 - 9, 2015, Princeton University, Mechanical and Aerospace Engineering, New York, USA, October 8, 2015.

  • A. Mielke, Mathematical modeling for finite-strain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 16, 2015.

  • A. Mielke, The Chemical Master Equation as entropic gradient flow, Conference ``New Trends in Optimal Transport'', March 2 - 6, 2015, Universität Bonn, Institut für Angewandte Mathematik, March 2, 2015.

  • A. Mielke, The Fokker--Planck and Liouville equations for chemical reactions as large-volume approximations of the Chemical Master Equation, Workshop ``Stochastic Limit Analysis for Reacting Particle Systems'', December 16 - 18, 2015, WIAS Berlin, Berlin, December 18, 2015.

  • A. Mielke, The multiplicative strain decomposition in finite-strain elastoplasticity, Intensive Period on Variational Methods for Plasticity and Dislocations, March 16 - 20, 2015, International School of Advanced Studies (SISSA), Trieste, Italy, March 17, 2015.

  • H. Neidhardt, Trace formulas for non-additive perturbations, Conference ``Spectral Theory and Applications'', May 25 - 29, 2015, AGH University of Science and Technology, Krakow, Poland, May 28, 2015.

  • J. Rehberg, On maximal parabolic regularity, The Fourth Najman Conference on Spectral Problems for Operators and Matrices, September 20 - 25, 2015, University of Zagreb, Department of Mathematics, Opatija, Croatia, September 23, 2015.

  • J. Rehberg, On maximal parabolic regularity and its applications, Oberseminar ``Mathematische Optimierung'', Technische Universität München, Lehrstuhl für Optimalsteuerung, München, May 6, 2015.

  • K. Disser, Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid, Second Workshop of the GAMM Activity Group on "Analysis of Partial Differential Equations", September 29 - October 1, 2014, Universität Stuttgart, Lehrstuhl für Analysis und Modellierung, October 1, 2014.

  • K. Disser, Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid, Autumn School and Workshop on Mathematical Fluid Dynamics, October 27 - 30, 2014, Universität Darmstadt, International Research Training Group 1529, Bad Boll, October 28, 2014.

  • K. Disser, Parabolic equations with mixed boundary conditions, degenerate diffusion and diffusion on Lipschitz interfaces, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Operator Theory, March 10 - 14, 2014, Friedrich-Alexander Universität Erlangen-Nürnberg, March 14, 2014.

  • K. Disser, Parabolic equations with mixed boundary conditions, degenerate diffusion and diffusion on Lipschitz interfaces, International Conference ``Vorticity, Rotations and Symmetry (III)---Approaching Limiting Cases of Fluid Flow'', May 5 - 9, 2014, Centre International de Rencontres Mathématiques (CIRM), Luminy, Marseille, France.

  • C. Kreisbeck, Thin-film limits of functionals on A-free vector fields and applications, Workshop on Trends in Non-Linear Analysis 2014, July 31 - August 1, 2014, Instituto Superior Técnico, Departamento de Matemática, Lisbon, Portugal, August 1, 2014.

  • C. Kreisbeck, Thin-film limits of functionals on A-free vector fields and applications, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8 - 11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

  • C. Kreisbeck, Thin-film limits of functionals on A-free vector fields and applications, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, July 16, 2014.

  • S. Reichelt, Effective equations for reaction-diffusion systems in strongly heterogeneous media, 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems (MURPHYS-HSFS-2014), April 7 - 11, 2014, WIAS Berlin, April 10, 2014.

  • S. Reichelt, Two-scale homogenization of nonlinear reaction-diffusion systems involving different diffusion length scales, scshape Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, December 3, 2014.

  • S. Reichelt, Two-scale homogenization of reaction-diffusion systems with small diffusion, 13th GAMM Seminar on Microstructures, January 17 - 18, 2014, Ruhr-Universität Bochum, Lehrstuhl für Mechanik - Materialtheorie, January 18, 2014.

  • S. Reichelt, Two-scale homogenization of reaction-diffusion systems with small diffusion, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 8: Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations from Mathematical Science, July 7 - 11, 2014, Madrid, Spain, July 8, 2014.

  • E. Rocca, ``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage, Symposium on Trends in Application of Mathematics to Mechanics (STAMM), September 8 - 11, 2014, International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 9, 2014.

  • S. Heinz, Analysis and numerics of a phase-transformation model, 13th GAMM Seminar on Microstructures, January 17 - 18, 2014, Ruhr-Universität Bochum, Lehrstuhl für Mechanik - Materialtheorie, January 18, 2014.

  • M. Liero, On dissipation distances for reaction-diffusion equations --- The Hellinger--Kantorovich distance, Workshop ``Entropy Methods, PDEs, Functional Inequalities, and Applications'', June 30 - July 4, 2014, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, July 1, 2014.

  • M. Liero, On dissipation distances for reaction-diffusion equations --- The Hellinger--Kantorovich distance, RIPE60 -- Rate Independent Processes and Evolution Workshop, June 24 - 26, 2014, Prague, Czech Republic, June 24, 2014.

  • S. Neukamm, Optimal quantitative two-scale expansion in stochastic homogenization, 13th GAMM Seminar on Microstructures, January 17 - 18, 2014, Ruhr-Universität Bochum, Lehrstuhl für Mechanik - Materialtheorie, January 18, 2014.

  • D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Conference on Nonlinearity, Transport, Physics, and Patterns, October 6 - 10, 2014, Fields Institute for Research in Mathematical Sciences, Toronto, Canada, October 9, 2014.

  • D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Oberseminar ``Stochastische und Geometrische Analysis'', Universität Bonn, Institut für Angewandte Mathematik, May 28, 2014.

  • E. Valdinoci, (Non)local interfaces and minimal surfaces, International Conference on ``Nonlinear Phenomena in Biology'', March 5 - 7, 2014, Helmholtz Zentrum München -- Deutsches Forschungszentrum für Gesundheit und Umwelt, March 5, 2014.

  • E. Valdinoci, Concentrating solutions for a nonlocal Schroedinger equation, Nonlinear Partial Differential Equations and Stochastic Methods, June 7 - 11, 2014, University of Jyväskylä, Finland, June 10, 2014.

  • E. Valdinoci, Concentration phenomena for nonlocal equation, Méthodes Géométriques et Variationnelles pour des EDPs Non-linéaires, September 1 - 5, 2014, Université C. Bernard, Lyon 1, Institut C. Jordan, France, September 2, 2014.

  • E. Valdinoci, Concentration solutions for a nonlocal Schroedinger equation, Kinetics, Non Standard Diffusion and the Mathematics of Networks: Emerging Challenges in the Sciences, May 7 - 16, 2014, The University of Texas at Austin, Department of Mathematics, USA, May 14, 2014.

  • E. Valdinoci, Dislocation dynamics and fractional equations, Analysis Seminar, Heriot-Watt University of Edinburgh, London, UK, October 31, 2014.

  • E. Valdinoci, Dislocation dynamics and fractional equations, Analysis Seminar, University of Texas at Austin Mathematics, USA, November 5, 2014.

  • E. Valdinoci, Dislocation dynamics in crystals, Recent Advances in Non-local and Non-linear Analysis: Theory and Applications, June 10 - 14, 2014, FIM -- Institute for Mathematical Research, ETH Zuerich, Switzerland, June 13, 2014.

  • E. Valdinoci, Dislocation dynamics in crystals, Geometry and Analysis Seminar, Columbia University, Department of Mathematics, New York City, USA, April 3, 2014.

  • E. Valdinoci, Dislocation dynamics in crystals, Seminari di Analisi Matematica, Università di Torino, Dipartimento di Matematica ``Giuseppe Peano'', Italy, December 18, 2014.

  • E. Valdinoci, Gradient estimates and symmetry results in anisotropic media, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 76: Viscosity, Nonlinearity and Maximum Principle, July 7 - 11, 2014, Madrid, Spain, July 8, 2014.

  • E. Valdinoci, Nonlinear PDEs, Spring School on Nonlinear PDEs, March 24 - 27, 2014, INdAM Istituto Nazionale d'Alta Matematica, Sapienza -- Università di Roma, Italy.

  • E. Valdinoci, Nonlocal equations and applications, Seminario de Ecuaciones Diferenciales, Universidad de Granada, IEMath-Granada, Spain, November 28, 2014.

  • E. Valdinoci, Nonlocal minimal surfaces, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 96: Geometric Variational Problems with Associated Stability Estimates, July 7 - 11, 2014, Madrid, Spain, July 8, 2014.

  • E. Valdinoci, Nonlocal minimal surfaces and free boundary problems, Geometric Aspects of Semilinear Elliptic and Parabolic Equations: Recent Advances and Future Perspectives, May 25 - 30, 2014, Banff International Research Station for Mathematical Innovation and Discovery, Calgary, Canada, May 27, 2014.

  • E. Valdinoci, Nonlocal problems in analysis and geometry, December 1 - 5, 2014, Universidad Autonoma de Madrid, Departamento de Matemáticas, Spain.

  • E. Valdinoci, Some nonlocal aspects of partial differential equations and free boundary problems, Institutskolloquium, Weierstrass Institut Berlin (WIAS), January 13, 2014.

  • A. Fiebach, A. Glitzky, Uniform estimate of the relative free energy by the dissipation rate for finite volume discretized reaction-diffusion systems, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), Berlin, June 15 - 20, 2014.

  • A. Glitzky, Drift-diffusion models for heterostructures in photovoltaics, 8th European Conference on Elliptic and Parabolic Problems, Minisymposium ``Qualitative Properties of Nonlinear Elliptic and Parabolic Equations'', May 26 - 30, 2014, Universität Zürich, Institut für Mathematik, organized in Gaeta, Italy, May 27, 2014.

  • D. Knees, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Analysis & Stochastics Seminar, Technische Universität Dresden, Institut für Analysis, January 16, 2014.

  • M. Thomas, A stress-driven local-solution approach to quasistatic brittle delamination, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 91: Variational Methods for Evolution Equations, July 7 - 11, 2014, Madrid, Spain, July 7, 2014.

  • M. Thomas, Existence & stability results for rate-independent processes in viscoelastic materials, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, March 18, 2014.

  • M. Thomas, Existence and stability results for rate-independent processes in viscoelastic materials, Women in Partial Differential Equations & Calculus of Variations Workshop, March 6 - 8, 2014, University of Oxford, Mathematical Institute, UK, March 6, 2014.

  • M. Thomas, GENERIC for solids with dissipative interface processes, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), GAMM Juniors' Poster Session, Friedrich-Alexander Universität Erlangen-Nürnberg, March 10 - 14, 2014.

  • M. Thomas, Rate-independent systems with viscosity and inertia: Existence and evolutionary Gamma-convergence, Workshop ``Variational Methods for Evolution'', December 14 - 20, 2014, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2014.

  • M. Thomas, Rate-independent, partial damage in thermo-viscoelastic materials, 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems (MURPHYS-HSFS-2014), April 7 - 11, 2014, WIAS Berlin, April 8, 2014.

  • M. Thomas, Rate-independent, partial damage in thermo-viscoelastic materials with inertia, International Workshop ``Variational Modeling in Solid Mechanics'', September 22 - 24, 2014, University of Udine, Department of Mathematics and Informatics, Italy, September 23, 2014.

  • M. Thomas, Rate-independent, partial damage in thermo-viscoelastic materials with inertia, Oberseminar ``Analysis und Angewandte Mathematik'', Universität Kassel, Institut für Mathematik, December 1, 2014.

  • M. Thomas, Stress-driven local-solution approach to quasistatic brittle delamination, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Analysis, March 10 - 14, 2014, Friedrich-Alexander Universität Erlangen-Nürnberg, March 11, 2014.

  • B. Wagner, Asymptotic analysis of interfacial evolution, BMS-WIAS Summer School ``Applied Analysis for Materials'', August 25 - September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

  • A. Mielke, S. Reichelt, M. Thomas, Pattern formation in systems with multiple scales, Evaluation of the DFG Collaborative Research Center 910 ``Control of Self-organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Berlin, June 30 - July 1, 2014.

  • A. Mielke, Evolutionary Gamma convergence and amplitude equations, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Analysis, March 10 - 14, 2014, Friedrich-Alexander Universität Erlangen-Nürnberg, March 13, 2014.

  • A. Mielke, Generalized gradient structures for reaction-diffusion systems, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, June 17, 2014.

  • A. Mielke, Gradient structures and dissipation distances for reaction-diffusion systems, Workshop ``Advances in Nonlinear PDEs: Analysis, Numerics, Stochastics, Applications'', June 2 - 3, 2014, Vienna University of Technology and University of Vienna, Austria, June 2, 2014.

  • A. Mielke, Gradient structures and dissipation distances for reaction-diffusion systems, Seminar ``Analysis of Fluids and Related Topics'', Princeton University, Department of Mechanical and Aerospace Engineering, Princeton, NJ, USA, March 6, 2014.

  • A. Mielke, Homogenization of parabolic gradient systems via evolutionary $Gamma$-convergence, Second Workshop of the GAMM Activity Group on ``Analysis of Partial Differential Equations'', September 29 - October 1, 2014, Universität Stuttgart, Institut für Analysis, Dynamik und Modellierung, September 30, 2014.

  • A. Mielke, How thermodynamics induces geometry structures for reaction-diffusion systems, Gemeinsames Mathematisches Kolloquium der Universitäten Gießen und Marburg, Universität Gießen, Mathematisches Institut, January 15, 2014.

  • A. Mielke, Multiscale modeling and evolutionary Gamma-convergence for gradient flows, BMS-WIAS Summer School ``Applied Analysis for Materials'', August 25 - September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

  • A. Mielke, On a metric and geometric approach to reaction-diffusion systems as gradient systems, Mathematics Colloquium, Jacobs University Bremen, School of Engineering and Science, December 1, 2014.

  • A. Mielke, On gradient structures and dissipation distances for reaction-diffusion systems, Kolloquium ``Angewandte Mathematik'', Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, July 3, 2014.

  • A. Mielke, On gradient structures for reaction-diffusion systems, Joint Analysis Seminar, Rheinisch-Westfälische Technische Hochschule Aachen (RWTH), Institut für Mathematik, February 4, 2014.

  • A. Mielke, On the microscopic origin of generalized gradient structures for reaction-diffusion systems, XIX International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2014), September 8 - 11, 2014, The International Society for the Interaction of Mechanics and Mathematics (ISIMM), Poitiers, France, September 11, 2014.

  • A. Mielke, A reaction-diffusion equation as a Hellinger--Kantorovich gradient flow, ERC Workshop on Optimal Transportation and Applications, October 27 - 31, 2014, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy, October 29, 2014.

  • S. Neukamm, Characterization and approximation of macroscopic properties in elasticity with homogenization, 4th British-German Frontiers of Science Symposium, Potsdam, March 6 - 9, 2014.

  • S. Neukamm, Characterization and approximation of macroscopic properties with homogenization, 4th British-German Frontiers of Science Symposium, March 6 - 9, 2014, Alexander von Humboldt-Stiftung, Potsdam, March 7, 2014.

  • S. Neukamm, Homogenization of nonlinear bending plates, Workshop ``Relaxation, Homogenization, and Dimensional Reduction in Hyperelasticity'', March 25 - 27, 2014, Université Paris-Nord, France, March 26, 2014.

  • S. Neukamm, Homogenization of slender structures in small-strain regimes, 14th Dresden Polymer Discussion, Meißen, May 25 - 28, 2014.

  • J. Rehberg, Maximal parabolic regularity on strange geometries and applications, Joint Meeting 2014 of the German Mathematical Society (DMV) and the Polish Mathematical Society (PTM), September 17 - 20, 2014, Adam Mickiewicz University, Faculty of Mathematics and Computer Science, Poznan, Poland, September 18, 2014.

  • J. Rehberg, On non-smooth parabolic equations, Workshop ``Maxwell--Stefan meets Navier--Stokes/Modeling and Analysis of Reactive Multi-Component Flows'', March 31 - April 2, 2014, Universität Halle, April 1, 2014.

  • J. Rehberg, Optimal Sobolev regularity for second order divergence operators, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Operator Theory, March 10 - 14, 2014, Friedrich-Alexander Universität Erlangen-Nürnberg, March 13, 2014.

  • K. Disser, Entropic gradient structures for reversible Markov chains and the passage to Wasserstein Fokker--Planck, Workshop for the Initiation of the GAMM Activity Group ``Analysis of Partial Differential Equations'', Regensburg, October 1 - 2, 2013.

  • K. Disser, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, BMS Intensive Course on Evolution Equations and their Applications, November 27 - 29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

  • K. Disser, Passage to the limit of the entropic gradient structure of reversible Markov processes to the Wasserstein Fokker--Planck equation, Oberseminar Analysis, Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Halle, November 20, 2013.

  • K. Götze, Free fall of a rigid body in a viscoelastic fluid, Workshop ``Geophysical Fluid Dynamics'', February 18 - 22, 2013, Mathematisches Forschungsinstitut Oberwolfach, February 20, 2013.

  • K. Götze, Maximal regularity for fluid-rigid body interaction problems, Interim Evaluation of the International Research Training Network 1529 ``Mathematical Fluid Dynamics'', Technische Universität Darmstadt, Fachbereich Mathematik, January 11, 2013.

  • K. Götze, Starke Lösungen für die Bewegung von Starrkörpern in Flüssigkeiten, Oberseminar Analysis, Technische Universität Dresden, Institut für Analysis, January 24, 2013.

  • S. Reichelt, Homogenization of degenerated reaction-diffusion equations, Doktorandenforum der Leibniz-Gemeinschaft, Sektion D, Berlin, June 6 - 7, 2013.

  • S. Reichelt, Introduction to homogenization concepts, Freie Universität Berlin, Institut für Mathematik, April 11, 2013.

  • S. Reichelt, Two-scale homogenization of nonlinear reaction-diffusion systems with small diffusion, Workshop on Control of Self-Organizing Nonlinear Systems, August 28 - 30, 2013, SFB 910 ``Control of self-organizing nonlinear systems: Theoretical methods and concepts of application'', Lutherstadt Wittenberg, August 30, 2013.

  • S. Reichelt, Two-scale homogenization in nonlinear reaction-diffusion systems with small diffusion, BMS Intensive Course on Evolution Equations and their Applications, November 27 - 29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

  • P. Gussmann, Linearized elasticity as $Gamma$-limit of finite elasticity in the case of cracks, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Section ``Applied Analysis'', March 18 - 22, 2013, University of Novi Sad, Serbia, March 20, 2013.

  • CH. Heinemann, Analysis of degenerating Cahn--Hilliard systems coupled with complete damage processes, 2013 CNA Summer School, Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, USA, May 30 - June 7, 2013.

  • CH. Heinemann, Degenerating Cahn--Hilliard systems coupled with complete damage processes, DIMO2013 -- Diffuse Interface Models, Levico Terme, Italy, September 10 - 13, 2013.

  • CH. Heinemann, Degenerating Cahn--Hilliard systems coupled with mechanical effects and complete damage processes, Equadiff13, MS27 -- Recent Results in Continuum and Fracture Mechanics, August 26 - 30, 2013, Prague, Czech Republic, August 27, 2013.

  • CH. Heinemann, On a PDE system describing damage processes and phase separation, Oberseminar Analysis, Universität Augsburg, July 11, 2013.

  • S. Heinz, On a way to control oscillations for a special evolution equation, Conference ``Nonlinearities'', June 10 - 14, 2013, University of Warsaw, Institute of Mathematics, Male Ciche, Poland, June 11, 2013.

  • M. Liero, Gradient structures and geodesic convexity for reaction-diffusion system, SIAM Conference on Mathematical Aspects of Materials Science (MS13), Minisymposium ``Material Modelling and Gradient Flows'' (MS100), June 9 - 12, 2013, Philadelphia, USA, June 12, 2013.

  • M. Liero, On gradient structures and geodesic convexity for reaction-diffusion systems, Research Seminar, Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik, April 17, 2013.

  • M. Liero, On gradient structures for drift-reaction-diffusion systems and Markov chains, Analysis Seminar, University of Bath, Mathematical Sciences, UK, November 21, 2013.

  • S. Neukamm, Quantitative results in stochastic homogenization, sc Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, June 27, 2013.

  • S. Neukamm, Quantitative results in stochastic homogenization, Oberseminar Analysis, Technische Universität Dresden, Fakultät Mathematik und Naturwissenschaften, June 13, 2013.

  • S. Neukamm, Optimal decay estimate on the semigroup associated with a random walk among random conductances, Dirichlet Forms and Applications, German-Japanese Meeting on Stochastic Analysis, September 9 - 13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 2013.

  • H. Abels, J. Daube, Ch. Kraus, D. Kröner, Sharp interface limit for the Navier--Stokes--Korteweg model, DIMO2013 -- Diffuse Interface Models, Levico Terme, Italy, September 10 - 13, 2013.

  • A. Glitzky, Continuous and finite volume discretized reaction-diffusion systems in heterostructures, Asymptotic Behaviour of Systems of PDE Arising in Physics and Biology: Theoretical and Numerical Points of View, November 6 - 8, 2013, Lille 1 University -- Science and Technology, France, November 6, 2013.

  • A. Glitzky, Nonlinear electrothermal feedback in organic semiconductors, Organic Photovoltaics Workshop 2013, December 10 - 11, 2013, University of Oxford, Mathematical Insitute, UK, December 10, 2013.

  • D. Knees, A vanishing viscosity approach to a rate-independent damage model, Seminar ``Wissenschaftliches Rechnen'', Technische Universität Dortmund, Fachbereich Mathematik, January 31, 2013.

  • D. Knees, Crack evolution models based on the Griffith criterion, Workshop on Mathematical Aspects of Continuum Mechanics, October 12 - 14, 2013, The Japan Society for Industrial and Applied Mathematics, Kanazawa, Japan, October 13, 2013.

  • D. Knees, Global spatial regularity for elasticity models with cracks and contact, Journées Singulières Augmentées 2013, August 26 - 30, 2013, Université de Rennes 1, France, August 27, 2013.

  • D. Knees, Global spatial regularity results for crack with contact and application to a fracture evolution model, Oberseminar Nichtlineare Analysis, Universität Köln, Mathematisches Institut, October 28, 2013.

  • D. Knees, Modeling and analysis of crack evolution based on the Griffith criterion, Nonlinear Analysis Seminar, Keio University of Science, Yokohama, Japan, October 9, 2013.

  • D. Knees, On energy release rates for nonlinearly elastic materials, Workshop on Mathematical Aspects of Continuum Mechanics, October 12 - 14, 2013, The Japan Society for Industrial and Applied Mathematics, Kanazawa, Japan, October 12, 2013.

  • D. Knees, Recent results in nonlinear elasticity and fracture mechanics, Universität der Bundeswehr, Institut für Mathematik und Bauinformatik, München, August 13, 2013.

  • D. Knees, Weak solutions for rate-independent systems illustrated at an example for crack propagation, BMS Intensive Course on Evolution Equations and their Applications, November 27 - 29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 28, 2013.

  • CH. Kraus, Damage and phase separation processes: Modeling and analysis of nonlinear PDE systems, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 11, 2013.

  • CH. Kraus, Modeling and analysis of a nonlinear PDE system for phase separation and damage, Università di Pavia, Dipartimento di Matematica, Italy, January 22, 2013.

  • CH. Kraus, Sharp interface limit of a diffuse interface model of Navier--Stokes--Allen--Cahn type for mixtures, Workshop ``Hyperbolic Techniques for Phase Dynamics'', June 10 - 14, 2013, Mathematisches Forschungsinstitut Oberwolfach, June 11, 2013.

  • M. Thomas, Damage and delamination processes in thermo-viscoelastic materials, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Young Reserchers' Minisymposium ``Analytical and Engineering Aspects in the Material Modeling of Solids'', March 18 - 22, 2013, University of Novi Sad, Serbia, March 19, 2013.

  • M. Thomas, Existence & fine properties of solutions for rate-independent brittle damage models, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM Juniors Poster Exhibition, Novi Sad, Serbia, March 18 - 22, 2013.

  • M. Thomas, Fine properties of solutions for rate-independent brittle damage models, XXIII Convegno Nazionale di Calcolo delle Variazioni, Levico Terme, Italy, February 3 - 8, 2013.

  • M. Thomas, Local versus energetic solutions in rate-independent brittle delamination, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 13, 2013.

  • M. Thomas, Rate-independent damage models with spatial BV-regularization -- Existence & fine properties of solutions, Oberseminar zur Analysis, Universität Duisburg-Essen, Fachbereich Mathematik, Essen, January 24, 2013.

  • M. Thomas, A stress-driven local solution approach to quasistatic brittle delamination, BMS Intensive Course on Evolution Equations and their Applications, November 27 - 29, 2013, Technische Universität Berlin, Berlin Mathematical School, November 29, 2013.

  • M. Thomas, A stress-driven local solution approach to quasistatic brittle delamination, Seminar on Functional Analysis and Applications, International School of Advanced Studies (SISSA), Trieste, Italy, November 12, 2013.

  • M. Thomas, Existence & fine properties of solutions for rate-independent brittle damage models, Workshop for the Initiation of the GAMM Activity Group ``Analysis of Partial Differential Equations'', Regensburg, October 1 - 2, 2013.

  • H. Hanke, Derivation of an effective damage model with evolving micro-structure, Oberseminar zur Analysis, Universität Duisburg-Essen, Fachbereich Mathematik, Essen, January 29, 2013.

  • H. Hanke, Derivation of an effective damage model with non-periodic evolving micro-structure, 12th GAMM Seminar on Microstructures, February 8 - 9, 2013, Humboldt-Universität zu Berlin, Institut für Mathematik, February 9, 2013.

  • A. Mielke, Gradient structures and dissipation distances for reaction-diffusion systems, Workshop ``Material Theory'', December 16 - 20, 2013, Mathematisches Forschungsinstitut Oberwolfach, December 17, 2013.

  • A. Mielke, Analysis, modeling, and simulation of semiconductor devices, Kolloquium Simulation Technology, Universität Stuttgart, SRC Simulation Technology, May 14, 2013.

  • A. Mielke, Deriving the Ginzburg--Landau equation as amplitude equation via evolutionary Gamma convergence, ERC Workshop on Variational Views on Mechanics and Materials, June 24 - 26, 2013, University of Pavia, Department of Mathematics, Italy, June 26, 2013.

  • A. Mielke, Evolutionary Gamma convergence and amplitude equations, sc Matheon Multiscale Seminar, Technische Universität Berlin, Institut für Mathematik, April 8, 2013.

  • A. Mielke, Gradient structures and uniform global decay for reaction-diffusion systems, Mathematisches Kolloquium, Universität Bielefeld, Fakultät für Mathematik, April 25, 2013.

  • A. Mielke, On entropy-driven dissipative quantum mechanical systems, Analysis and Stochastics in Complex Physical Systems, March 20 - 22, 2013, Universität Leipzig, Mathematisches Institut, March 21, 2013.

  • A. Mielke, On the geometry of reaction-diffusion systems: Optimal transport versus reaction, Recent Trends in Differential Equations: Analysis and Discretisation Methods, November 7 - 9, 2013, Technische Universität Berlin, Institut für Mathematik, November 9, 2013.

  • A. Mielke, Rate-independent plasticity as vanishing-viscosity limit for wiggly energy landscape, Workshop on Evolution Problems for Material Defects: Dislocations, Plasticity, and Fracture, September 30 - October 4, 2013, International School of Advanced Studies (SISSA), Trieste, Italy, September 30, 2013.

  • A. Mielke, Using gradient structures for modeling semiconductors, Eindhoven University of Technology, Institute for Complex Molecular Systems, Netherlands, February 21, 2013.

  • J. Rehberg, Sobolev extention and analysis on non-Lipschitz domain, Oberseminar Angewandte Analysis, Universität Darmstadt, Fachbereich Mathematik, May 14, 2013.

  • J. Sprekels, Optimal control of Allen--Cahn equations with singular potentials and dynamic boundary conditions, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 11, 2013.

  • J. Sprekels, Optimal control of the Allen--Cahn equation with dynamic boundary condition and double obstacle potentials: A ``deep quench'' approach, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, September 17, 2013.

  • J. Sprekels, Prandtl--Ishlinskii operators and elastoplasticity, Spring School on ``Rate-independent Evolutions and Hysteresis Modelling'', May 27 - 31, 2013, Politecnico di Milano, Università degli Studi di Milano, Italy.

  • K. Götze, Free fall of a rigid body in a viscoelastic fluid, International Summer School on Evolution Equations (EVEQ 2012), Prague, Czech Republic, July 9 - 13, 2012.

  • K. Götze, Free fall of a rigid body in a viscoelastic fluid, International Summer School on Evolution Equations (EVEQ 2012), July 9 - 13, 2012, Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

  • K. Götze, Some results for the motion of a rigid body with a liquid-filled cavity, Seminar on Partial Differential Equations, Czech Academy of Sciences, Mathematical Institute, Prague, Czech Republic, November 6, 2012.

  • CH. Heinemann, Complete damage in linear elastic materials, Variational Models and Methods for Evolution, Levico, Italy, September 10 - 12, 2012.

  • CH. Heinemann, Damage processes coupled with phase separation in elastically stressed alloys, GAMM Jahrestagung 2012 (83rd Annual Meeting), March 26 - 30, 2012, Technische Universität Darmstadt, March 27, 2012.

  • CH. Heinemann, Existence of weak solutions for rate-dependent complete damage processes, Materialmodellierungsseminar, WIAS, Berlin, October 31, 2012.

  • CH. Heinemann, Kopplung von Phasenseparation und Schädigung in elastischen Materialien, Leibniz-Doktoranden-Forum der Sektion D, Berlin, June 7 - 8, 2012.

  • S. Heinz, Quasiconvexity equals rank-one convexity for isotropic sets of 2x2 matrices, 11th GAMM Seminar on Microstructures, January 20 - 21, 2012, Universität Duisburg-Essen, January 20, 2012.

  • S. Heinz, Regularization and relaxation of time-continuous problems in plasticity, 11th GAMM Seminar on Microstructures, Universität Duisburg-Essen, January 20 - 21, 2012.

  • S. Heinz, Rigorous derivation of a dissipation for laminate microstructures, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Material Modelling in Solid Mechanics, March 26 - 30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

  • M. Liero, Interfaces in reaction-diffusion systems, Seminar ``Dünne Schichten'', Technische Universität Berlin, Institut für Mathematik, February 9, 2012.

  • M. Liero, Variational methods for evolution, ``A sc Matheon Multiscale Workshop'', Technische Universität Berlin, Institut für Mathematik, April 20, 2012.

  • A. Glitzky, An electronic model for solar cells taking into account active interfaces, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24 - 28, 2012, WIAS Berlin, September 27, 2012.

  • A. Glitzky, Mathematische Modellierung und Simulation organischer Halbleiterbauelemente, Senatsausschuss Wettbewerb (SAW), Sektion D der Leibniz-Gemeinschaft, Leibniz-Institut für Analytische Wissenschaften (ISAS), Dortmund, September 14, 2012.

  • D. Knees, A vanishing viscosity approach in fracture mechanics, Nonlocal Models and Peridynamics, November 5 - 7, 2012, Technische Universität Berlin, Institut für Mathematik, November 5, 2012.

  • D. Knees, Modeling and mathematical analysis of elasto-plastic phenomena, Winter School on Modeling Complex Physical Systems with Nonlinear (S)PDE (DoM$^2$oS), February 27 - March 2, 2012, Technische Universität Dortmund, Fakultät für Mathematik.

  • D. Knees, On a vanishing viscosity approach in damage mechanics, Kolloquium der AG Modellierung, Numerik, Differentialgleichungen, Technische Universität Berlin, Institut für Mathematik, August 21, 2012.

  • CH. Kraus, A nonlinear PDE system for phase separation and damage, Universität Freiburg, Abteilung Angewandte Mathematik, November 13, 2012.

  • CH. Kraus, Cahn--Larché systems coupled with damage, Università degli Studi di Milano, Dipartimento di Matematica, Italy, November 28, 2012.

  • CH. Kraus, Phase field systems for phase separation and damage processes, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11 - 15, 2012, Frauenchiemsee, June 12, 2012.

  • CH. Kraus, Phasenfeldsysteme für Entmischungs- und Schädigungsprozesse, Mathematisches Kolloquium, Universität Stuttgart, Fachbereich Mathematik, May 15, 2012.

  • CH. Kraus, The Stefan problem with inhomogeneous and anisotropic Gibbs--Thomson law, 6th European Congress of Mathematics, July 2 - 6, 2012, Cracow, Poland, July 5, 2012.

  • M. Thomas, A model for rate-independent, brittle delamination in thermo-visco-elasticity, International Workshop on Evolution Problems in Damage, Plasticity, and Fracture: Mathematical Models and Numerical Analysis, September 19 - 21, 2012, University of Udine, Department of Mathematics, Italy, September 21, 2012.

  • M. Thomas, A model for rate-independent, brittle delamination in thermo-visco-elasticity, INDAM Workshop PDEs for Multiphase Advanced Materials (ADMAT2012), September 17 - 21, 2012, Cortona, Italy, September 17, 2012.

  • M. Thomas, Analytical aspects of rate-independent damage models with spatial BV-regularization, Seminar, SISSA -- International School for Advanced Studies, Functional Analysis and Applications, Trieste, Italy, November 28, 2012.

  • M. Thomas, Delamination in visco-elastic materials with thermal effects, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Applied Analysis, March 26 - 30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

  • M. Thomas, Delamination in viscoelastic materials with thermal effects, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11 - 15, 2012, Universität Regensburg, Frauenchiemsee, June 11, 2012.

  • M. Thomas, Delamination in viscoelastic materials with thermal effects, Seminar on Applied Mathematics, Università di Brescia, Dipartimento di Matematica, Italy, March 14, 2012.

  • M. Thomas, Mathematical methods in continuum mechanics of solids, COMMAS (Computational Mechanics of Materials and Structures) Summer School, October 8 - 12, 2012, Universität Stuttgart, Institut für Mechanik (Bauwesen).

  • M. Thomas, Modellierung und Analysis von Delaminationsprozessen, Sitzung des Wissenschaftlichen Beirats des WIAS, Berlin, October 5, 2012.

  • M. Thomas, Rate-independent evolution of sets, Variational Models and Methods for Evolution, Levico, Italy, September 10 - 12, 2012.

  • M. Thomas, Thermomechanical modeling via energy and entropy using GENERIC, Workshop ``Mechanics of Materials'', March 19 - 23, 2012, Mathematisches Forschungsinstitut Oberwolfach, March 22, 2012.

  • A. Fiebach, A. Glitzky, A. Linke, Voronoi finite-volume methods for reaction-diffusion-systems, 33. Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik (NoKo 2012), Universität Rostock, Institut für Mathematik, May 4 - 5, 2012.

  • H. Hanke, Derivation of an effective damage evolution model, ``A sc Matheon Multiscale Workshop'', Technische Universität Berlin, Institut für Mathematik, April 20, 2012.

  • H. Hanke, Derivation of an effective damage evolution model using two-scale convergence techniques, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Damage Processes and Contact Problems, March 26 - 30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 29, 2012.

  • H. Hanke, Derivation of an effective damage evolution model using two-scale convergence techniques, International Workshop on Evolution Problems in Damage, Plasticity, and Fracture: Mathematical Models and Numerical Analysis, September 19 - 21, 2012, University of Udine, Department of Mathematics, Italy, September 19, 2012.

  • A. Mielke, Entropy gradient flows for Markow chains and reaction-diffusion systems, Berlin-Leipzig-Seminar ``Analysis/Probability Theory'', WIAS Berlin, April 13, 2012.

  • A. Mielke, Finite-strain viscoelasticity as a gradient flow, Analysis and Applications of PDEs: An 80th Birthday Meeting for Robin Knops, December 10 - 11, 2012, International Center for Mathematical Sciences, Edinburgh, UK, December 11, 2012.

  • A. Mielke, Gamma convergence and evolution, International Conference ``Trends in Mathematical Analysis'', March 1 - 3, 2012, Politecnico di Milano, Dipartimento di Matematica ``F. Brioschi'', Italy, March 1, 2012.

  • A. Mielke, Gradienten-Strukturen und geodätische Konvexität für Markov-Ketten und Reaktions-Diffusions-Systeme, Augsburger Kolloquium, Universität Augsburg, Institut für Mathematik, May 8, 2012.

  • A. Mielke, Linearized elastoplasticity is the evolutionary Gamma limit of finite elastoplasticity, 83th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2012), Session on Applied Analysis, March 26 - 30, 2012, Technische Universität Darmstadt, Fachbereich Mathematik, March 27, 2012.

  • A. Mielke, Multidimensional modeling and simulation of optoelectronic devices, Challenge Workshop ``Modeling, Simulation and Optimisation Tools'', September 24 - 26, 2012, Technische Universität Berlin, September 24, 2012.

  • A. Mielke, Multiscale modeling for evolutionary systems via Gamma convergence, NDNS$^+$ Summer School in Applied Analysis, June 18 - 20, 2012, University of Twente, Applied Analysis & Mathematical Physics, Enschede, Netherlands.

  • A. Mielke, On consistent couplings of quantum mechanical and dissipative systems, Jahrestagung der Deutschen Mathematiker-Vereinigung (DMV) 2012, Minisymposium ``Dynamical Systems'', September 17 - 20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 19, 2012.

  • A. Mielke, On geodesic convexity for reaction-diffusion systems, Seminar on Applied Mathematics, Università di Pavia, Dipartimento di Matematica, Italy, March 6, 2012.

  • A. Mielke, On gradient structures and geodesic convexity for energy-reaction-diffusion systems and Markov chains, ERC Workshop on Optimal Transportation and Applications, November 5 - 9, 2012, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy, November 8, 2012.

  • A. Mielke, On gradient structures for Markov chains and reaction-diffusion systems, Applied & Computational Analysis (ACA) Seminar, University of Cambridge, Department of Applied Mathematics and Theoretical Physics (DAMTP), UK, June 14, 2012.

  • A. Mielke, Using gradient structures for modeling semiconductors, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24 - 28, 2012, WIAS Berlin, September 24, 2012.

  • J. Rehberg, Hölderstetigkeit für Lösungen elliptischer Gleichungen, Seminar der Arbeitsgruppe ``Analysis'', Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, December 12, 2012.

  • J.A. Griepentrog, On nonlocal phase separation processes in multicomponent systems, 10th GAMM Seminar on Microstructures, January 20 - 22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 22, 2011.

  • S. Heinz, A model for the texture evolution in polycrystalline materials, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Solid Mechanics, April 18 - 21, 2011, Technische Universität Graz, Austria, April 19, 2011.

  • A. Petrov, Vibrations with unilateral constraints: An overview of M. Schatzman's contributions - Part II: Deformable bodies, 7th International Congress on Industrial and Applied Mathematics, Minisymposium ``Vibrations with Unilateral Constraints'', July 18 - 22, 2011, Society for Industrial and Applied Mathematics, Vancouver, Canada, July 22, 2011.

  • J.A. Griepentrog, The role of nonsmooth regularity theory in the analysis of phase separation processes, Ehrenkolloquium anlässlich des 60. Geburtstages von PD Dr. habil. Lutz Recke, Humboldt-Universität zu Berlin, Institut für Mathematik, November 21, 2011.

  • S. Heinz, Regularizations and relaxations of time-continuous problems in plasticity, Workshop der Forschergruppe 797 ``Analysis and Computation of Microstructure in Finite Plasticity'', Universität Bonn, Mathematisches Institut, November 14, 2011.

  • U. Stefanelli, Evolution = Minimization?, Friday Colloquium, Berlin Mathematical School, May 27, 2011.

  • K. Götze, Starke Lösungen für die Interaktion von starren Körpern und viskoelastischen Flüssigkeiten, Lectures in Continuum Mechanics, Universität Kassel, Institut für Mathematik, November 7, 2011.

  • A. Glitzky, An electronic model for solar cells including active interfaces, Workshop ``Mathematical Modelling of Organic Photovoltaic Devices'', University of Cambridge, Department of Applied Mathematics and Theoretical Physics, UK, June 9, 2011.

  • CH. Heinemann, Existence results for Cahn-Hilliard equations coupled with elasticity and damage, Workshop on Phase Separation, Damage and Fracture, September 21 - 23, 2011, WIAS, September 23, 2011.

  • D. Knees, A vanishing viscosity approach in damage mechanics, Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), June 20 - 24, 2011, Gargnano (Brescia), Italy, June 22, 2011.

  • D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, Workshop ``Perspectives in Continuum Mechanics'' in Honor of Gianfranco Capriz's 85th Birthday, University of Florence, Department of Mathematics, Italy, January 28, 2011.

  • D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, 7th International Congress on Industrial and Applied Mathematics, Session ``Materials Science II'', July 18 - 22, 2011, Society for Industrial and Applied Mathematics, Vancouver, Canada, July 19, 2011.

  • D. Knees, On a vanishing viscosity approach for a model in damage mechanics, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Applied Analysis, April 18 - 21, 2011, Technische Universität Graz, Austria, April 20, 2011.

  • CH. Kraus, Diffuse interface systems for phase separation and damage, Seminar on Partial Differential Equations, Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, May 3, 2011.

  • CH. Kraus, Phase separation systems coupled with elasticity and damage, ICIAM 2011, July 18 - 22, 2011, Vancouver, Canada, July 18, 2011.

  • A. Mielke, Complex hysteresis operators arising from homogenization and dimension reduction, 8th International Symposium on Hysteresis Modelling and Micromagnetics (HMM2011), Session ``Mathematics of Hysteresis II'', May 9 - 11, 2011, Università degli Studi di Trento, Centro Internazionale per la Ricerca Matematica, Levico Terme, Italy, May 10, 2011.

  • A. Mielke, Multiscale problems in systems driven by functionals, ISAM-TopMath Summer School 2011 on Variational Methods, September 12 - 16, 2011, Technische Universität München, Fakultät für Mathematik.

  • A. Mielke, Remarks on evolutionary multiscale systems driven by functionals, Intellectual Challenges in Multiscale Modeling of Solids, July 4 - 5, 2011, Oxford University, Mathematical Institute, UK, July 4, 2011.

  • J. Sprekels, A nonstandard phase field system of Cahn--Hilliard type for diffusive phase segregation, Seminario Matematico e Fisico di Milano, Università degli Studi di Milano, Dipartimento di Matematica, Italy, September 21, 2011.

  • M. Thomas, Modeling and analysis of rate-independent damage and delamination processes, 19th International Conference on Computer Methods in Mechanics, Minisymposium ``Growth Phenomena and Evolution of Microstructures. Applications in Solids'', May 9 - 12, 2011, Warsaw University of Technology, Poland, May 11, 2011.

  • D. Knees, A vanishing viscosity approach in damage mechanics, Workshop ``Variational Methods for Evolution'', December 5 - 10, 2011, Mathematisches Forschungsinstitut Oberwolfach, December 5, 2011.

  • D. Knees, Analysis und Numerik für quasistatische Rissausbreitung, Lectures in Continuum Mechanics, Universität Kassel, Fachbereich für Mathematik und Naturwissenschaften, December 12, 2011.

  • A. Mielke, Mathematical approaches to thermodynamic modeling, Autumn School on Mathematical Principles for and Advances in Continuum Mechanics, November 7 - 12, 2011, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

  • J. Sprekels, Phase field models and hysteresis operators, Trends in Thermodynamics and Materials Theory 2011, December 15 - 17, 2011, Technische Universität Berlin, December 16, 2011.

  • J. Sprekels, Well-posedness, asymptotic behavior and optimal control of a nonstandard phase field model for diffusive phase segregation, Workshop on Optimal Control of Partial Differential Equations, November 28 - December 1, 2011, Wasserschloss Klaffenbach, Chemnitz, November 30, 2011.

  • M. Liero, Rate-independent Kurzweil processes, Workshop ``Rate-independent Systems: Modeling, Analysis, and Computations'', August 30 - September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, September 3, 2010.

  • A. Petrov, On a 3D model for shape-memory alloys, Workshop ``Rate-independent Systems: Modeling, Analysis, and Computations'', August 30 - September 3, 2010, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, September 2, 2010.

  • A. Petrov, Viscoelastodynamic problem with Signorini boundary conditions, 81th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2010), Session on Applied Analysis, March 22 - 26, 2010, Universität Karlsruhe, March 25, 2010.

  • A. Glitzky, Existence of bounded steady state solutions to spin-polarized drift-diffusion systems, Workshop on Drift Diffusion Systems and Related Problems: Analysis, Algorithms and Computations, WIAS, Research Group ``Numerical Mathematics and Scientific Computing'', March 25, 2010.

  • A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Special Session on Reaction Diffusion Systems, May 25 - 28, 2010, Technische Universität Dresden, May 26, 2010.

  • CH. Kraus, An inhomogeneous, anisotropic and elastically modified Gibbs-Thomson law as singular limit of a diffuse interface model, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), March 22 - 26, 2010, Karlsruhe, March 23, 2010.

  • CH. Kraus, Inhomogeneous and anisotropic phase-field quantities in the sharp interface limit, 6th Singular Days 2010, April 29 - May 1, 2010, WIAS, Berlin, April 30, 2010.

  • A. Mielke, Approaches to finite-strain elastoplasticity, SIAM Conference on Mathematical Aspects of Materials Science (MS10), May 23 - 26, 2010, Philadelphia, USA, May 23, 2010.

  • A. Mielke, Geometrische Nichtlinearitäten und Lie-Gruppen in der Elastoplastizität, Mathematisches Kolloquium, Technische Universität Darmstadt, Fachbereich Mathematik, January 27, 2010.

  • A. Mielke, Gradient structures for electro-reaction-diffusion systems with applications in photovoltaics, First Interdisciplinary Workshop of the German-Russian Interdisciplinary Science Center (G-RISC) ``Structure and Dynamics of Matter'', October 18 - 20, 2010, Freie Universität Berlin and Helmholtz-Zentrum Berlin für Materialien und Energie, October 19, 2010.

  • A. Mielke, Gradient structures for reaction-diffusion systems and semiconductor models with interface dynamics, International Conference on Evolution Equations, October 11 - 14, 2010, Technische Universität Darmstadt, Fachbereich Mathematik, Schmitten, October 12, 2010.

  • J. Rehberg, On parabolic equations with non-homogeneous Neumann boundary conditions, International Conference on Evolution Equations, October 11 - 14, 2010, Technische Universität Darmstadt, Fachbereich Mathematik, Schmitten, October 11, 2010.

  • J. Rehberg, Optimal elliptic regularity near 3-dimensional, heterogeneous Neumann vertices, 6th Singular Days on Asymptotic Methods for PDEs, April 29 - May 1, 2010, WIAS, April 29, 2010.

  • J. Sprekels, Technical and mathematical problems in the Czochralski growth of single crystals, Workshop ``New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries'', January 31 - February 6, 2010, Mathematisches Forschungsinstitut Oberwolfach, February 1, 2010.

  • H. Stephan, Asymptotisches Verhalten positiver Halbgruppen, Oberseminar Analysis, Technische Universität Dresden, Institut für Analysis, January 14, 2010.

  • H. Stephan, Evolution equations conserving positivity, Colloquium of Centre for Analysis, Scientific Computing and Applications (CASA), Technische Universiteit Einhoven, Netherlands, April 21, 2010.

  • J.A. Griepentrog, Maximal regularity for nonsmooth parabolic boundary value problems in Sobolev--Morrey spaces, International Conference on Elliptic and Parabolic Equations, November 30 - December 4, 2009, WIAS, December 1, 2009.

  • S. Heinz, A model for the evolution of laminates, 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2009), Young Researchers Minisymposium ``Mathematics and Mechanics of Microstructure Evolution in Finite Plasticity'', February 9 - 13, 2009, Gdansk University of Technology, Poland, February 10, 2009.

  • S. Heinz, The evolution of laminates, 8th GAMM Seminar on Microstructures, January 15 - 17, 2009, Universität Regensburg, NWF-I Mathematik, January 17, 2009.

  • A. Petrov, On existence for viscoelastodymanic problems with unilateral boundary conditions, 80th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2009), Session ``Applied analysis'', February 9 - 13, 2009, Gdansk University of Technology, Poland, February 10, 2009.

  • A. Petrov, On the error estimates for space-time discretizations of rate-independent processes, 8th GAMM Seminar on Microstructures, January 15 - 17, 2009, Universität Regensburg, NWF-I Mathematik, January 17, 2009.

  • A. Petrov, On the existence and error bounds for space-time discretizations of a 3D model for shape-memory alloys, Lisbon University, Center for Mathematics and Fundamental Applications, Portugal, September 17, 2009.

  • A. Petrov, On the numerical approximation of a viscoelastic problem with unilateral constrains, 7th EUROMECH Solid Mechanics Conference (ESMC2009), Minisymposium on Contact Mechanics, September 7 - 11, 2009, Instituto Superior Técnico, Lisbon, Portugal, September 8, 2009.

  • W. Dreyer, Stochastic evolution of many particle storage systems, 9th Hirschegg Workshop on Conservation Laws, September 6 - 12, 2009, Hirschegg, September 10, 2009.

  • J. Fuhrmann, Mathematical and numerical modeling of fuel cells and electrochemical flow cells, Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, Kaiserslautern, December 1, 2009.

  • J. Fuhrmann, Modeling of reaction diffusion problems in semiconductor photoresists, Seventh Negev Applied Mathematical Workshop, July 6 - 8, 2009, Ben Gurion University of the Negev, Jacob Blaustein Institute for Desert Research, Sede Boqer Campus, Israel, July 8, 2009.

  • A. Glitzky, Discrete Sobolev--Poincaré inequalities for finite volume Voronoi approximations, Annual Meeting of the Deutsche Mathematiker-Vereinigung and 17th Congress of the Österreichische Mathematische Gesellschaft, Section ``Partial Differential Equations'', September 20 - 25, 2009, Technische Universität Graz, Austria, September 21, 2009.

  • A. Glitzky, Discrete Sobolev--Poincaré inequalities using the $W^1,p$ seminorm in the setting of Voronoi finite volume approximations, International Conference on Elliptic and Parabolic Equations, November 30 - December 4, 2009, WIAS, December 3, 2009.

  • H.-Chr. Kaiser, Transient Kohn--Sham theory, Jubiläumssymposium ``Licht -- Materialien -- Modelle'' (100 Jahre Innovation aus Adlershof), Berlin-Adlershof, September 7 - 8, 2009.

  • H. Neidhardt, Linear non-autonomous Cauchy problems and evolution semigroups, International Bogolyubov Conference ``Problems of Theoretical and Mathematical Physics'', August 21 - 27, 2009, Moscow State University/Joint Institute for Nuclear Research, Dubna, Russian Federation, August 24, 2009.

  • J. Polzehl, Sequential multiscale procedures for adaptive estimation, The 1st Institute of Mathematical Statistics Asia Pacific Rim Meeting, June 28 - July 1, 2009, Seoul National University, Institute of Mathematical Statistics, Korea (Republic of), July 1, 2009.

  • J. Rehberg, Functional analytic properties of the quantum mechanical particle density operator, International Workshop on Quantum Systems and Semiconductor Devices: Analysis, Simulations, Applications, April 20 - 24, 2009, Peking University, School of Mathematical Sciences, Beijing, China, April 21, 2009.

  • J. Rehberg, Quasilinear parabolic equations in distribution spaces, International Conference on Nonlinear Parabolic Problems in Honor of Herbert Amann, May 10 - 16, 2009, Stefan Banach International Mathematical Center, Bedlewo, Poland, May 12, 2009.

  • H. Stephan, Inequalities for Markov operators, Positivity VI (Sixth Edition of the International Conference on Positivity and its Applications), July 20 - 24, 2009, El Escorial, Madrid, Spain, July 24, 2009.

  • H. Stephan, Modeling of diffusion prozesses with hidden degrees of freedom, Workshop on Numerical Methods for Applications, November 5 - 6, 2009, Lanke, November 6, 2009.

  • A. Petrov, Error estimates for space-time discretizations of a 3D model for shape-memory materials, IUTAM Symposium ``Variational Concepts with Applications to the Mechanics of Materials'', September 22 - 26, 2008, Ruhr-Universität Bochum, Lehrstuhl für allgemeine Mechanik, September 24, 2008.

  • A. Petrov, Existence and approximation for 3D model of thermally induced phase transformation in shape-memory alloys, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), Session ``Material models in solids'', March 31 - April 4, 2008, Universität Bremen, April 1, 2008.

  • A. Petrov, Some mathematical results for a model of thermally-induced phase transformations in shape-memory materials, sc Matheon--ICM Workshop on Free Boundaries and Materials Modeling, March 17 - 18, 2008, WIAS, March 18, 2008.

  • H.-Chr. Kaiser, A drift-diffusion model for semiconductors with internal interfaces, Annual Meeting of the Deutsche Mathematiker-Vereinigung 2008, Minisymposium ``Analysis of Reaction-Diffusion Systems with Internal Interfaces'', September 15 - 19, 2008, Friedrich-Alexander-Universität Erlangen-Nürnberg, September 15, 2008.

  • H.-Chr. Kaiser, A thermodynamic approach to transient Kohn--Sham theory, 100th Statistical Mechanics Conference, December 13 - 18, 2008, Rutgers, The State University of New Jersey, New Brunswick, USA, December 16, 2008.

  • H.-Chr. Kaiser, On drift-diffusion Kohn--Sham theory, 79th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2008), session ``Applied Analysis'', March 31 - April 4, 2008, University of Bremen, April 1, 2008.

  • CH. Kraus, A phase-field model with anisotropic surface tension in the sharp interface limit, Second GAMM-Seminar on Multiscale Material Modelling, July 10 - 12, 2008, Universität Stuttgart, Institut für Mechanik (Bauwesen), July 12, 2008.

  • CH. Kraus, Ein Phasenfeldmodell vom Cahn-Hilliard-Typ im singulären Grenzwert, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, April 25, 2008.

  • CH. Kraus, Phase field models and corresponding Gibbs--Thomson laws. Part II, SIMTECH Seminar Multiscale Modelling in Fluid Mechanics, Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation, November 5, 2008.

  • J. Rehberg, Hölder continuity for elliptic and parabolic problems, Analysis-Tag, Technische Universität Darmstadt, Fachbereich Mathematik, November 27, 2008.

  • A. Petrov, On the convergence for kinetic variational inequality to quasi-static variational inequality with application to elastic-plastic systems with hardening, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16 - 20, 2007, ETH Zürich, Switzerland, July 17, 2007.

  • A. Petrov, Thermally driven phase transformation in shape-memory alloys, Workshop ``Analysis and Numerics of Rate-Independent Processes'', February 26 - March 2, 2007, Mathematisches Forschungsinstitut Oberwolfach, February 27, 2007.

  • A. Glitzky, Energy estimates for reaction-diffusion processes of charged species, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16 - 20, 2007, ETH Zürich, Switzerland, July 16, 2007.

  • H.-Chr. Kaiser, A drift-diffusion model of transient Kohn--Sham theory, First Joint International Meeting between the American Mathematical Society and the Polish Mathematical Society, Special Session ``Mathematics of Large Quantum Systems'', July 31 - August 3, 2007, University of Warsaw, Poland, August 3, 2007.

  • CH. Kraus, On jump conditions at phase interfaces, Oberseminar über Angewandte Mathematik, December 10 - 15, 2007, Universität Freiburg, Abteilung für Angewandte Mathematik, December 11, 2007.

  • J. Rehberg, An elliptic model problem including mixed boundary conditions and material heterogeneities, Fifth Singular Days, April 23 - 27, 2007, International Center for Mathematical Meetings, Luminy, France, April 26, 2007.

  • J. Rehberg, Maximal parabolic regularity on Sobolev spaces, The Eighteenth Crimean Autumn Mathematical School-Symposium (KROMSH-2007), September 17 - 29, 2007, Laspi-Batiliman, Ukraine, September 18, 2007.

  • F. Schmid, An evolution model in contact mechanics with dry friction, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16 - 20, 2007, ETH Zürich, Switzerland, July 19, 2007.

  • A. Petrov, Mathematical result on the stability of elastic-plastic systems with hardening, European Conference on Smart Systems, October 26 - 28, 2006, Researching Training Network "New Materials, Adaptive Systems and their Nonlinearities: Modelling, Control and Numerical Simulation" within the European Commission's 5th Framework Programme, Rome, Italy, October 27, 2006.

  • CH. Kraus, Equilibrium conditions for liquid-vapor system in the sharp interface limit, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, July 18, 2006.

  • A. Glitzky, Energy estimates for electro-reaction-diffusion systems with partly fast kinetics, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25 - 28, 2006, Université de Poitiers, France, June 27, 2006.

  • A. Glitzky, Energy models where the equations are defined on different domains, GAMM Annual Meeting 2006, March 27 - 31, 2006, Technische Universität Berlin, March 29, 2006.

  • CH. Kraus, Equilibria conditions in the sharp interface limit of the van der Waals-Cahn-Hilliard phase model, Recent Advances in Free Boundary Problems and Related Topics (FBP2006), September 14 - 16, 2006, Levico, Italy, September 14, 2006.

  • CH. Kraus, The sharp interface limit of the van der Waals--Cahn--Hilliard model, Polish-German Workshop ``Modeling Structure Formation'', Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Poland, September 8, 2006.

  • H. Neidhardt, Classical solutions of van Roosbroeck's semiconductor equations, Conference ``Mathematical aspects of quantum transport in mesoscopic systems'', December 18 - 20, 2006, Université de Toulon et Centre de Physique Théorique, Marseille, France, December 18, 2006.

  • H. Neidhardt, Perturbation theory of semigroups and evolution equations, Jahrestagung der Deutschen Mathematiker-Vereinigung 2006, September 18 - 21, 2006, Universität Bonn, September 20, 2006.

  • J. Rehberg, Existence and uniqueness for van Roosbroeck's system in Lebesque spaces, Conference ``Recent Advances in Nonlinear Partial Differential Equations and Applications'', Toledo, Spain, June 7 - 10, 2006.

  • J. Rehberg, Regularity for nonsmooth elliptic problems, Crimean Autumn Mathematical School, September 20 - 25, 2006, Vernadskiy Tavricheskiy National University, Laspi, Ukraine, September 21, 2006.

  • CH. Kraus, On the sharp limit of the Van der Waals-Cahn-Hilliard model, WIAS Workshop ``Dynamic of Phase Transitions'', November 30 - December 3, 2005, Berlin, December 2, 2005.

  • CH. Kraus, On the sharp limit of the Van der Waals-Cahn-Hilliard model, Workshop ``Micro-Macro Modeling and Simulation of Liquid-Vapor Flows'', November 16 - 18, 2005, Universität Freiburg, Mathematisches Institut, Kirchzarten, November 17, 2005.

  • CH. Kraus, Maximale Konvergenz in höheren Dimensionen, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, May 24, 2005.

  • J. Rehberg, Elliptische und parabolische Probleme aus Anwendungen, Kolloquium im Fachbereich Mathematik, Universität Darmstadt, May 18, 2005.

  • J. Rehberg, Existence, uniqeness and regularity for quasilinear parabolic systems, International Conference ``Nonlinear Partial Differential Equations'', September 17 - 24, 2005, Institute of Applied Mathematics and Mechanics Donetsk, Alushta, Ukraine, September 18, 2005.

  • J. Rehberg, H$^1,q$-regularity for linear, elliptic boundary value problems, Regularity for nonlinear and linear PDEs in nonsmooth domains - Analysis, simulation and application, September 5 - 7, 2005, Universität Stuttgart, Deutsche Forschungsgemeinschaft (SFB 404), Hirschegg, Austria, September 6, 2005.

  • J. Rehberg, Regularität für elliptische Probleme mit unglatten Daten, Oberseminar Prof. Escher/Prof. Schrohe, Technische Universität Hannover, December 13, 2005.

  • J. Rehberg, Some analytical ideas concerning the quantum-drift-diffusion systems, Workshop ``Problèmes spectraux non-linéaires et modèles de champs moyens'', April 4 - 8, 2005, Institut Henri Poincaré, Paris, France, April 5, 2005.

  • J. Rehberg, Analysis of macroscopic and quantum mechanical semiconductor models, International Visitor Program ``Nonlinear Parabolic Problems'', August 8 - November 18, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, November 1, 2005.

  • J. Rehberg, Existence, uniqueness and regularity for quasilinear parabolic systems, Conference ``Nonlinear Parabolic Problems'', October 17 - 21, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, October 20, 2005.

  • J. Rehberg, Elliptische und parabolische Probleme mit unglatten Daten, Technische Universität Darmstadt, Fachbereich Mathematik, December 14, 2004.

  • J. Rehberg, Quasilinear parabolic equations in $L^p$, Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann, June 28 - 30, 2004, Universität Zürich, Institut für Mathematik, Switzerland, June 29, 2004.

  • J. Rehberg, The two-dimensional van Roosbroeck system has solutions in $L^p$, Workshop ``Advances in Mathematical Semiconductor Modelling: Devices and Circuits'', March 2 - 6, 2004, Chinese-German Centre for Science Promotion, Beijing, China, March 5, 2004.

  • U. Bandelow, Thermodynamic designed energy model, 3rd Topical meeting on Numerical Simulation of Semiconductor Optoelectronic Devices (NUSOD'03), October 14 - 16, 2003, University of Tokyo, Japan, October 14, 2003.

  • H.-Chr. Kaiser, Classical solutions of van Roosbroeck's equations with discontinuous coefficients and mixed boundary conditions on two-dimensional space domains, 19th GAMM Seminar Leipzig on High-dimensional problems --- Numerical treatment and applications, January 23 - 25, 2003, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 25, 2003.

  • J. Rehberg, A combined quantum mechanical and macroscopic model for semiconductors, Workshop on Multiscale problems in quantum mechanics and averaging techniques, December 11 - 12, 2003, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, December 12, 2003.

  • J. Rehberg, Solvability and regularity for parabolic equations with nonsmooth data, International Conference ``Nonlinear Partial Differential Equations'', September 15 - 21, 2003, National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics, Alushta, September 17, 2003.

  External Preprints

  • A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Preprint no. arXiv:1510.06219, Cornell University Library, arXiv.org, 2015.
    Abstract
    Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

  • D.A. Gomes, S. Patrizi, Obstacle mean-field game problem, Preprint no. arXiv:1410.6942, Cornell University Library, arXiv.org, 2014.
    Abstract
    In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.

  • S. Neukamm, A. Gloria, F. Otto, An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, Preprint no. 41, Max-Planck-Institut für Mathematik in den Naturwissenschaften, 2013.
    Abstract
    We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Greens function by Marahrens and the third author.

  • A. Fiaschi, D. Knees, U. Stefanelli, Young measure quasi-static damage evolution, Preprint no. 28PV10/26/0, Istituto di Matematica Applicata e Technologie Informatiche, 2010.

  • A. Mielke, On an evolutionary model for complete damage based on energies and stresses, Preprint no. 29PV10/27/0, pp. 23--32, in: Rate-independent evolutions and material modeling, T. Roubíček, U. Stefanelli (eds.), Istituto di Matematica Applicata e Technologie Informatiche, 2010.

  • D. Knees, Griffith-formula and J-integral for a crack in a power-law hardening material, Preprint no. 2005/12, Universität Stuttgart, SFB 404, 2005.

  • M. Lichtner, M. Radziunas, Well posedness and smooth dependence for a semilinear hyperbolic system with nonsmooth data, Preprint no. 174, DFG Research Center sc Matheon, Technische Universität Berlin, 2004.