Extremal principles play a fundamental role in physics and mechanics. They state that a system adjusts its state always in a manner such that the associated functional is extremal: A soap bubble minimizes the surface area subject to a given volume, an elastic body minimizes the stored elastic energy subject to given boundary conditions, and a temperaturedependent system maximizes entropy subject to a given energy. Thus, one can determine equilibrium states in a canonical way by studying local or global minimizers or maximizers of a functional.
Moreover, many nonequilibrium phenomena exhibit a variational structure where some energy functional is minimized, for example in Lagrangian mechanics or in gradient flows. A deeper understanding of these variational structures unveals the workings behind the dynamics, which can be used for scale bridging purposes.
Applications of variational methods are found in many areas:
 linear and nonlinear elastostatics and contact problems
 microstructures in plasticity and shape memory alloys
 free boundary problems and multiphase problems
 equilibria in reactiondiffusion systems
 ground states in quantum mechanics and density functional theory
 optimization
 risk minimization in finance and economy
Derivation of effective multiscale models
At WIAS, the strong tools provided by the calculus of variations are further improved and used in the treatment of partial differential equations in all of the above applications. Especially in the field of multiscale modeling, e.g. for dimension reduction or for the characterization of microstructures in solids, variational techniques such as Gammaconvergence are extended and applied to obtain novel effective models that offer for example an easier numerical treatment. Moreover, such techniques often turn out to be inevitable for the understanding of the transition from discrete systems to rescaled continuous approximations.
Variational methods for evolution problems
While the direct methods of calculus of variations deal with minimization problems, i.e. static problems, at WIAS we aim at extending the rich toolbox of calculus of variations also to evolutionary problems. In particular, we consider systems that are driven by functionals. Generalized gradient systems, for example, are systems whose evolution is driven by an energy or entropy functional towards equilibrium while a convex dissipation potential controls the dissipated energy or entropy during this evolution. Classical gradient systems correspond to quadratic dissipation potentials, however, also non quadratic dissipation potentials are of great interest. In rateindependent systems for instance the dissipation potentials are positively homogeneous of degree one. In Mielke, Peletier, Renger it was shown that discrete Markovian jump processes lead via large deviation principles to generalized gradient systems with nonquadratic dissipation potentials.
Large deviation principles
Variational problems also serve as the basis of a natural formulation of exponential decay rates of certain probabilities in the light of the theory of large deviations, which is used at WIAS to make the asymptotic description of models of statistical mechanics explicit. They usually reveal deeper properties and structures of the model, and their minimizers describe optimal (i.e., most probable) configurations of the system.
Highlights
Gradient structures for reactiondiffusion and energydriftdiffusion equations
Starting from the pioneering work of Felix Otto, the deep connection between optimal transport problems and scalar diffusion problems has been under intensive research in the last decades. One of the essential connecting factors is the interpretation of the KantorovichWasserstein distance in a Riemannian sense, i.e. it is induced by a symmetric and positive semidefinite differential operator, also called Onsager operator, whose inverse is an infinitedimensional version of a Riemannian tensor. Starting from this important observation, generalizations of the Onsager operator were introduced in Mielke 2011, that can be associated with certain PDEs of reactiondiffusiontype via gradientflow formulations. In particular, the energydriftdiffusion systems used in the modeling of charge and energy flow in semiconductor devices were shown to have this beneficial structure.
HellingerKantorovich distances and entropytransport problems
In Liero, Mielke, Savaré 2016 several useful characterizations of a new distance on the space of all finite, nonnegative measures were established. The distance is induced by an Onsager operator that leads to scalar reaction diffusion equation modeling the spreading and growth of biological tissue. It is called HellingerKantorovich distance as it can be seen as a convolution between the HellingerKakutani distance and the KantorovichWasserstein distance. The most interesting characterization of the distance gives rise to socalled entropytransport problems that consists of a classical transport part, describing the effective cost of moving some mass, and an entropy part penalizing the deviation of the initial and final transported mass distributions from the initial and the final given mass distributions. New fields of applications, for example in image analysis, emerge for this new class of entropytransport problems.Publications
Monographs

H.Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Science, Springfield, 2017, IV+933 pages, (Collection Published).
Abstract
HAGs von Christoph bestätigen lassen 
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 sharmonic 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 selfcontained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance. 
W. König, The Parabolic Anderson Model  Random Walks in Random Potential, Pathways in Mathematics, Birkhäuser, Basel, 2016, xi+192 pages, (Monograph Published).

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. 187249, (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 E_{ε} and the dissipation potential R_{ε} or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions u_{ε}. We investigate the question under which conditions the limits u of (subsequences of) the solutions u_{ε} are solutions of the gradient system generated by the Γlimits E_{0} and R_{0}. Here the choice of the right topology will be crucial as well as additional structural conditions.
We cover classical gradient systems, where R_{ε} is quadratic, and rateindependent systems as well as the passage from viscous to rateindependent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results. 
A. Mielke, T. Roubíček, Rateindependent 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 rateindependent systems, which the authors have been working on with a lot of collaborators over 15 years. The focus is mostly on fully rateindependent 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 adhesivetype contacts both at small strains and at finite strains. A few other physical systems, e.g. magnetic or ferroelectric materials, and couplings to ratedependent 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 "rateindependent 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. 125155, (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 Gammaconvergence. 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 rateindependent system. On the other hand we show how the concept of BalancedViscosity solution arise as in the vanishingviscosity limit.
As applications we discuss, first, the evolution of laminate microstructures in finitestrain elastoplasticity and, second, a twophase model for shapememory materials, where Hmeasures 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).

G. Dal Maso, A. Mielke, U. Stefanelli, eds., Rateindependent 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, Chapter: Differential, Energetic, and Metric Formulations for RateIndependent Processes, in: Nonlinear PDE's and Applications, C.I.M.E. Summer School, Cetraro, Italy 2008, L. Ambrosio, G. Savaré, eds., 2028 of Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 2011, pp. 87167, (Chapter Published).
Abstract
We consider different solution concepts for rateindependent systems. This includes energetic solutions in the topological setting and differentiable, local, parametrized and BV solutions in the Banachspace setting. The latter two solution concepts rely on the method of vanishing viscosity, in which solutions of the rateindependent system are defined as limits of solutions of systems with small viscosity. Finally, we also show how the theory of metric evolutionary systems can be used to define parametrized and BV solutions in metric spaces. 
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. 31453216, (Chapter Published).
Articles in Refereed Journals

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. 135, DOI 10.3934/dcdss.2017001 .
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 nonquadratic 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 largedeviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to coshtype dissipation potentials. A second origin arises via a new form of convergence, that we call EDPconvergence. 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 Gammalimit. 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 reactiondiffusion system. 
R. Rossi, M. Thomas, Coupling rateindependent and ratedependent processes: Existence results, SIAM Journal on Mathematical Analysis, 49 (2017), pp. 14191494.
Abstract
We address the analysis of an abstract system coupling a rateindependet 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 timediscretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rateindependent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rateindependent processes in viscoelastic solids with inertia, and to a recently proposed model for damage with plasticity. 
R. Rossi, M. Thomas, From adhesive to brittle delamination in viscoelastodynamics, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 14891546, DOI 10.1142/S0218202517500257 .
Abstract
In this paper we analyze a system for brittle delamination between two viscoelastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rateindependent 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 rateindependent systems to the present mixed ratedependent/rateindependent 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. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), pp. 25182546.
Abstract
In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by HawkinsDaruud et al. in citeHZO. The model consists of a CahnHilliard equation for the tumor cell fraction $vp$ coupled to a reactiondiffusion equation for a function $s$ representing the nutrientrich extracellular water volume fraction. The distributed control $u$ monitors as a righthand 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 controltostate operator is Fréchet differentiable between appropriate Banach spaces and derive the firstorder necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. 
P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Asymptotic analyses and error estimates for a CahnHilliard type phase field system modelling tumor growth, Discrete and Continuous Dynamical Systems, 10 (2017), pp. 3754.
Abstract
This paper is concerned with a phase field system of CahnHilliard 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 wellposedness, 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 
CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperaturedependent phasefield model for phase separation and damage, Archive for Rational Mechanics and Analysis, 225 (2017), pp. 177247.
Abstract
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 CahnHilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179211]), 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), 461490] in the framework of FourierNavierStokes 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), 13451369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 25192586] for the study of PDE systems for phase transition and damage. Our globalintime existence result is obtained by passing to the limit in a carefully devised timediscretization scheme. 
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. 7093.
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 firstorder necessary and secondorder 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. 
M. Thomas, Ch. Zanini, Cohesive zonetype delamination in viscoelasticity, Discrete and Continuous Dynamical Systems  Series S, 10 (2017), pp. 14871517, DOI 10.20347/WIAS.PREPRINT.2350 .
Abstract
We study a model for the rateindependent evolution of cohesive zone delamination in a viscoelastic 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 nonpenetration 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&Thomas15WIASPreprint2113]. 
A. Mielke, R. Rossi, G. Savaré, Global existence results for viscoplasticity at finite strain, Archive for Rational Mechanics and Analysis, (2017), published online on 20.9.2017, DOI 10.1007/s0020501711646 .
Abstract
We study a model for ratedependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of globalintime solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finitestrain elasticity as well as the multiplicative decomposition of finitestrain plasticity. Moreover, the dissipation potential depends on the leftinvariant plastic rate and thus, depends on the plastic state variable.
The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energydissipationbalance (EDB) and energydissipationinequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory. 
A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Nonequilibrium thermodynamical principles for chemical reactions with massaction kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 15621585, DOI 10.20347/WIAS.PREPRINT.2165 .
Abstract
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 nonlinear relation between thermodynamic fluxes and free energy driving force. 
E. Cinti, J. Davila, M. Del Pino, Solutions of the fractional AllenCahn equation which are invariant under screw motion, Journal of the London Mathematical Society. Second Series, 94 (2016), pp. 295313.
Abstract
We establish existence and nonexistence results for entire solutions to the fractional AllenCahn equation in R3 , which vanish on helicoids and are invariant under screwmotion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature. 
E. Cinti, F. Otto, Interpolation inequalities in pattern formation, Journal of Functional Analysis, 271 (2016), pp. 10431376.
Abstract
We prove some interpolation inequalities which arise in the analysis of pattern formation in physics. They are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts (coarsening and branching in micromagnetics and superconductors). The main ingredient in the proof of our inequalities is a geometric construction which was first used by Choksi, Conti, Kohn, and one of the authors in [4] in the study of branching in superconductors. 
M. Cozzi, A. Farina, E. Valdinoci, Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs, Advances in Mathematics, 293 (2016), pp. 343381.
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 nonsingular and non degenerate anisotropic equations. 
S.P. Frigeri, Global existence of weak solutions for a nonlocal model for twophase flows of incompressible fluids with unmatched densities, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 19571993.
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 NavierStokes type system coupled with a convective nonlocal CahnHilliard 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 doublewell 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. Liero, A. Mielke, G. Savaré, Optimal transport in competition with reaction: The HellingerKantorovich distance and geodesic curves, SIAM Journal on Mathematical Analysis, 48 (2016), pp. 28692911.
Abstract
We discuss a new notion of distance on the space of finite and nonnegative measures on Ω ⊂ ℝ ^{d}, which we call HellingerKantorovich distance. It can be seen as an infconvolution of the wellknown KantorovichWasserstein distance and the HellingerKakutani 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. 
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. 189210.
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 NavierStokes 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. 86/186/25.
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. 71/171/44.
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). 
A. Farina, E. Valdinoci, 1D symmetry for semilinear PDEs from the limit interface of the solution, Communications in Partial Differential Equations, 41 (2016), pp. 665682.
Abstract
We study bounded, monotone solutions of ?u = W?(u) in the whole of ?n, where W is a doublewell 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. 
X. RosOton, E. Valdinoci, The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains, Advances in Mathematics, 288 (2016), pp. 732790.
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 onedimensional 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. 
A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of NonEquilibrium Thermodynamics, 41 (2016), pp. 141149.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium 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 timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
A. Mielke, T. Roubíček, Rateindependent elastoplasticity at finite strains and its numerical approximation, Mathematical Models & Methods in Applied Sciences, 26 (2016), pp. 22032236.
Abstract
Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rateindependent evolution. The energy functional with a frameindifferent 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 nonselfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses 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 infinitedimensional rateindependent systems, Journal of the European Mathematical Society (JEMS), 18 (2016), pp. 21072165.
Abstract
Balanced Viscosity solutions to rateindependent systems arise as limits of regularized rateindependent ows by adding a superlinear vanishingviscosity dissipation. We address the main issue of proving the existence of such limits for innitedimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energydissipation identity. A careful description of the jump behavior of the solutions, of their dierentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chainrule inequality for functions of bounded variation in Banach spaces, on rened lower semicontinuitycompactness arguments, and on new BVestimates that are of independent interest. 
K. Disser, M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks and Heterogeneous Media, 10 (2015), pp. 233253.
Abstract
We study the approximation of Wasserstein gradient structures by their finitedimensional analog. We show that simple finitevolume discretizations of the linear FokkerPlanck equation exhibit the recently established entropic gradientflow 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 gradientflow structures. In particular, we make no use of the linearity of the equations nor of the fact that the FokkerPlanck equation is of second order. 
S. Patrizi, E. Valdinoci, Crystal dislocations with different orientations and collisions, Archive for Rational Mechanics and Analysis, 217 (2015), pp. 231261.
Abstract
We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of PeierlsNabarro 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. 336.
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 socalled 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. 25192586.
Abstract
In this paper we analyze a PDE system modelling (nonisothermal) 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 righthand 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 globalintime 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 timediscrete analysis could be useful towards the numerical study of this model. 
S.P. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for twophase incompressible flows with nonlocal interactions and nonconstant mobility, Nonlinearity, 28 (2015), pp. 12571293.
Abstract
We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the NavierStokes system coupled with a convective nonlocal CahnHilliard equation with nonconstant mobility. We first prove the existence of a global weak solution in the case of nondegenerate 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 CahnHilliard equation with degenerate mobility and singular potential in dimension three. 
D. Peschka, Thinfilm free boundary problems for partial wetting, Journal of Computational Physics, 295 (2015), pp. 770778.
Abstract
We present a novel framework to solve thinfilm equations with an explicit nonzero contact angle, where the support of the solution is treated as an unknown. The algorithm uses a finite element method based on a gradient formulation of the thinfilm equations coupled to an arbitrary LagrangianEulerian method for the motion of the support. Features of this algorithm are its simplicity and robustness. We apply this algorithm in 1D and 2D to problems with surface tension, contact angles and with gravity. 
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. 39283959.
Abstract
In this work, we analytically investigate a multicomponent 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. 24212464.

S. Dipierro, E. Valdinoci, On a fractional harmonic replacement, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 33773392.
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. 45594605.
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 thermoviscoelasticity, ESAIM. Control, Optimisation and Calculus of Variations, 21 (2015), pp. 159.
Abstract
We address the analysis of a model for brittle delamination of two viscoelastic 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 BrezisNirenberg result for the fractional Laplacian, Transactions of the American Mathematical Society, 367 (2015), pp. 67102.

J. Dávila, M. Del Pino, S. Dipierro, E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Analysis & PDE, 8 (2015), pp. 11651235.
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. Erbar, J. Maas, D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Electronic Communications in Probability, 20 (2015), pp. 112.
Abstract
We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of Gammaconvergence) to the JordanKinderlehrerOtto functional arising in the Wasserstein gradient flow structure of the FokkerPlanck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions. 
M.M. Fall, F. Mahmoudi, E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), pp. 19371961.
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 BallMajumdar 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; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 194 (2015), pp. 12691299.
Abstract
In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landaude Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible NavierStokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Qtensor $QQ$, namely a tensorvalued 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 Qtensor 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 blowup 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. 35513563.
Abstract
In this paper we study a nonlocal fractional Laplace equation, depending on a parameter, with asymptotically linear righthand 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 meanfield game problem, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 17 (2015), pp. 5568.
Abstract
In this paper, we introduce and study a firstorder meanfield game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and powerlike nonlinearities. Since the obstacle operator is not differentiable, the equations for firstorder 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. 
R. Huth, S. Jachalski, G. Kitavtsev, D. Peschka, Gradient flow perspective on thinfilm bilayer flows, Journal of Engineering Mathematics, 94 (2015), pp. 4361.
Abstract
We study gradient flow formulations of thinfilm bilayer flows with triplejunctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes freeboundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thinfilm model and formally identify solutions with those of the corresponding freeboundary problem. A robust numerical algorithm for the thinfilm gradient flow structure is then provided. Using this algorithm we compare the sharp triplejunction model with precursor models. For their stationary solutions a rigorous connection is established using Gammaconvergence. For timedependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zerocontact angle case, where we compare numerical solutions with asymptotically exact sourcetype 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. 555587.

T. Roubíček, M. Thomas, Ch. Panagiotopoulos, Stressdriven localsolution approach to quasistatic brittle delamination, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 645663.
Abstract
A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rateindependent 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 semiimplicit scheme and regularized by a BVtype gradient term. An analytical zerodimensional example motivates the model and a specific localsolution concept. Twodimensional 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. 
M. Thomas, Uniform PoincaréSobolev and relative isoperimetric inequalities for classes of domains, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 27412761.
Abstract
The aim of this paper is to prove an isoperimetric inequality relative to a ddimensional, 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 PoincaréSobolev 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). 
H. Hanke, D. Knees, Homogenization of elliptic systems with nonperiodic, state dependent coefficients, Asymptotic Analysis, 92 (2015), pp. 203234.
Abstract
In this paper, a homogenization problem for an elliptic system with nonperiodic, 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 nonperiodic 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 timedependent damage models (discussed in a future paper), we provide a Γconvergence 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 CahnHilliard system coupled with complete damage processes, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 22 (2015), pp. 388403.
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 timedepending domain for phase separation and complete damage processes under timevarying 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 quasistatic 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 onesided 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 nondegenerating 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. 217250.
Abstract
We introduce a complete damage model with a timedepending domain for linearelastically stressed solids under timevarying Dirichlet boundary conditions. The evolution of the system is described by a doubly nonlinear differential inclusion for the damage process and a quasistatic 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 localintime existence and globalintime 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 timedepending domain. In this context, two major challenges arise: Firstly, the timedependent 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 timedepending 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. 25652590.
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 hyperbolicparabolic 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, timediscretization and different variational techniques. 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for a hyperbolicparabolic phase field system with mixed boundary conditions on nonsmooth domains, SIAM Journal on Mathematical Analysis, 47 (2015), pp. 20442073.
Abstract
The aim of this paper is to prove existence of weak solutions of hyperbolicparabolic 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 timediscretization, $H^2$regularization and variational techniques. 
CH. Heinemann, E. Rocca, Damage processes in thermoviscoelastic materials with damagedependent thermal expansion coefficients, Mathematical Methods in the Applied Sciences, 38 (2015), pp. 45874612.
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 damagedependent thermal expansion coefficient. This term implies the presence of nonlinear couplings in the PDE system, which make the analysis more challenging. 
A. Mielke, Deriving amplitude equations via evolutionary Gamma convergence, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 26792700.
Abstract
We discuss the justification of the GinzburgLandau equation with real coefficients as an amplitude equation for the weakly unstable onedimensional SwiftHohenberg 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 timedependent 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 L^{2}, while for the case of a quadratic nonlinearity we need to impose weak convergence in H^{1}. However, we do not need wellpreparedness of the initial conditions. 
H. Stephan, Multiplikative Dualität in der Dreiecksgeometrie, Die Wurzel, Zeitschrift für Mathematik, 49 (2015), pp. 105110.

H. Stephan, Reverse inequalities for slowly increasing sequences and functions, Octogon Mathematical Magazine, 22 (2015), pp. 621633.
Abstract
We consider sharp inequalities involving slowly increasing sequences and functions, i.e., functions $f(t)$ with $f'(t) leq 1$ and sequences $(a_i)$ with $a_i+1a_i leq 1$. The inequalities are reverse to mean inequalities, for example. In the continuous case, integrals of powers are estimated by powers of integrals, whereas in the discrete case powers of sums are estimated by sums of powers of sums. The problem is connected with interpolation theory in Banach spaces, one of them $W^1,infty$. 
H. Stephan, Zahlentheorie und Geometrie, Mitteilungen der Mathematischen Gesellschaft in Hamburg, 35 (2015), pp. 1844.

C. Kreisbeck, L. Mascarenhas, Asymptotic spectral analysis in semiconductor nanowire heterostructures, Applicable Analysis. An International Journal, (published online on June 2, 2014), DOI 10.1080/00036811.2014.919052 .

E. Rocca, R. Rossi, A degenerating PDE system for phase transitions and damage, Mathematical Models & Methods in Applied Sciences, 24 (2014), pp. 12651341.

S. Heinz, On the structure of the quasiconvex hull in planar elasticity, Calculus of Variations and Partial Differential Equations, 50 (2014), pp. 481489.
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 rankone convex hull in planar elasticity. 
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. 629650.

A. Cesaroni, M. Novaga, E. Valdinoci, A symmetry result for the OrnsteinUhlenbeck operator, Discrete and Continuous Dynamical Systems, 34 (2014), pp. 24512467.

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. 831855.

N. Abatangelo, E. Valdinoci, A notion of nonlocal curvature, Numerical Functional Analysis and Optimization. An International Journal, 35 (2014), pp. 793815.

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. 189214.

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. 383404.

A. Farina, E. Valdinoci, Gradient bounds for anisotropic partial differential equations, Calculus of Variations and Partial Differential Equations, 49 (2014), pp. 923936.

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. 156170.

A. Gloria, S. Neukamm, F. Otto, An optimal quantitative twoscale expansion in stochastic homogenization of discrete elliptic equations, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), pp. 325346.
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 twoscale asymptotic expansion has the same scaling as in the periodic case. In particular the L^{2}norm in probability of the H^{1}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. 
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. 677699.

A. Miranville, E. Rocca, G. Schimperna, A. Segatti, The PenroseFife phasefield model with coupled dynamic boundary conditions, Discrete and Continuous Dynamical Systems, 34 (2014), pp. 42594290.

O. Savin, E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, Journal de Mathématiques Pures et Appliquées, 101 (2014), pp. 126.

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. 4979.

A. Mielke, Ch. Ortner, Y. Şengül, An approach to nonlinear viscoelasticity via metric gradient flows, SIAM Journal on Mathematical Analysis, 46 (2014), pp. 13171347.
Abstract
We formulate quasistatic nonlinear finitestrain viscoelasticity of ratetype as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure timedependent frameindifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the timeincremental 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 onedimensional 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 timecontinuous in some case in a specific case. 
A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the largedeviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), pp. 12931325.
Abstract
Motivated by the occurence in rate functions of timedependent largedeviation principles, we study a class of nonnegative functions ℒ that induce a flow, given by ℒ(z_{t},ż_{t})=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 entropyWasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure. 
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), DOI 10.1007/s0052601407652 .
Abstract
We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gammaconvergence. 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 twoscale convergence. This is a nontrivial task, since one has to treat twoscale convergence in connection with a nonlinear differential constraint. 
H. Stephan, Verallgemeinerungen der Jensenschen Ungleichung, Die Wurzel, Zeitschrift für Mathematik, 48 (2014), pp. 187194.

CH. Heinemann, Ch. Kraus, Existence results for diffuse interface models describing phase separation and damage, European Journal of Applied Mathematics, 24 (2013), pp. 179211.
Abstract
In this paper we analytically investigate CahnHilliard and AllenCahn systems which are coupled with elasticity and unidirectional 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 CahnHilliard and AllenCahn systems coupled with unidirectional 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. 
M. Liero, U. Stefanelli, A new minimum principle for Lagrangian mechanics, Journal of Nonlinear Science, 23 (2013), pp. 179204.
Abstract
We present a novel variational approach to Lagrangian mechanics based on elliptic regularization with respect to time. A class of parameterdependent globalintime 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 timehorizon version of this approach, and provide related Gammaconvergence results. Finally, some discussion on corresponding timediscrete versions of the principle is presented. 
M. Liero, U. Stefanelli, Weighted InertiaDissipationEnergy functionals for semilinear equations, Bollettino della Unione Matematica Italiana. Serie 9, VI (2013), pp. 127.

M. Liero, A. Mielke, Gradient structures and geodesic convexity for reactiondiffusion systems, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013), pp. 20120346/120120346/28.
Abstract
We consider systems of reactiondiffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a socalled Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambdaconvexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a driftdiffusion system, provide a survey on the applicability of the theory. We consider systems of reactiondiffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a socalled Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambdaconvexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a driftdiffusion system, provide a survey on the applicability of the theory. 
M. Liero, Passing from bulk to bulk/surface evolution in the AllenCahn equation, NoDEA. Nonlinear Differential Equations and Applications, 20 (2013), pp. 919942.
Abstract
In this paper we formulate a boundary layer approximation for an AllenCahntype 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^2gradient flow with the corresponding AllenCahn 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 vonKármán plate theory from 3D nonlinear elasticity, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 27012748.
Abstract
We rigorously derive a homogenized vonKármán plate theory as a Gammalimit from nonlinear threedimensional elasticity by combining homogenization and dimension reduction. Our starting point is an energy functional that describes a nonlinear elastic, threedimensional 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 vonKármán scaling regime. The obtained limit is a homogenized vonKármán plate model. Its effective material properties are determined by a relaxation formula that exposes a nontrivial coupling of the behavior of the outofplane displacement with the oscillatory behavior in the inplane 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. 269288.
Abstract
We consider integral functionals with densities of pgrowth, 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 pgrowth 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. 
A. Bradji, J. Fuhrmann, Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes, Applications of Mathematics, 58 (2013), pp. 138.
Abstract
A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems by R. Eymard and coworkers. Thanks to these basic ideas developed for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. Although the numerical scheme stems from the finite volume method, its formulation is based on the discrete version for the weak formulation defined for the heat problem. We derive error estimates for the solution in discrete norm, and an error estimate for an approximation of the gradient, in a general framework in which the discrete bilinear form is satisfying ellipticity. We prove in particular, that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is h+k , where h (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption that the exact solution is twice continuously differentiable in time and space. These error estimates are useful because they allow us to get error estimates for the approximations of the exact solution and its first derivatives. 
M.H. Duong, V. Laschos, M. Renger, Wasserstein gradient flows from large deviations of manyparticle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013), pp. 11661188.

C.P. Niculescu, H. Stephan, Lagrange's barycentric identity from an analytic viewpoint, Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie. Nouvelle Serie, 56 (104) (2013), pp. 487496.
Abstract
We discuss a generalization of Lagrange's algebraic identity that provides valuable insights into the nature of Jensen's inequality and of many other inequalities of convexity. 
M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the FokkerPlanck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/11350017/43.

A. Glitzky, A. Mielke, A gradient structure for systems coupling reactiondiffusion 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. 2952.
Abstract
We derive gradientflow formulations for systems describing driftdiffusion processes of a finite number of species which undergo massaction 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 electroreactiondiffusion systems with active interfaces permitting driftdiffusion 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 selfconsistent 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 rateindependent damage model, Mathematical Models & Methods in Applied Sciences, 23 (2013), pp. 565616.
Abstract
We analyze a rateindependent 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 bynow classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rateindependent damage models as limits of systems driven by viscous, ratedependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparameterization. In this way, in the limit we obtain a novel formulation for the rateindependent damage model, which highlights the interplay of viscous and rateindependent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps. 
M. Thomas, Quasistatic damage evolution with spatial BVregularization, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 235255.
Abstract
An existence result for energetic solutions of rateindependent 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∈ W^{1,r}(Ω) with r∈(1,∞) for Ω⊂R^{d}. We now cover the case r=1. The lack of compactness in W^{1,1}(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 Γlimit of functionals of ModicaMortola type. 
P.N. Racec, S. Schade, H.Chr. Kaiser, Eigensolutions of the WignerEisenbud problem for a cylindrical nanowire within finite volume method, Journal of Computational Physics, 252 (2013), pp. 5264.
Abstract
We present a finite volume method for computing a representative range of eigenvalues and eigenvectors of the Schrödinger operator on a three dimensional cylindrically symmetric bounded domain with mixed boundary conditions. More specifically, we deal with a semiconductor nanowire which consists of a dominant host material and contains heterostructure features such as doublebarriers or quantum dots. The three dimensional Schrödinger operator is reduced to a family of two dimensional Schrödinger operators distinguished by a centrifugal potential. Ultimately, we numerically treat them by means of a finite volume method. We consider a uniform, boundary conforming Delaunay mesh, which additionally conforms to the material interfaces. The 1/r singularity is eliminated by approximating r at the vertexes of the Voronoi boxes. We study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first K eigenvalues and eigenvectors and their sequential arrangement. There exists an optimal uniform Delaunay discretization with matching anisotropy. This anisotropic discretization yields best accuracy also in the presence of a mildly varying scattering potential, shown exemplarily for a nanowire resonant tunneling diode. For potentials with 1/r singularity one retrieves the theoretically established first order convergence, while the second order convergence is recovered only on uniform grids with an anisotropy correction. 
H. Stephan, Multiplicative duality in triangles, Recreatii Matematice, 15 (2013), pp. 1821.

K. Hackl, S. Heinz, A. Mielke, A model for the evolution of laminates in finitestrain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 92 (2012), pp. 888909.
Abstract
We study the time evolution in elastoplasticity within the rateindependent 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 timeincremental 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. 
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. 18591884.
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 FEdiscretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results. 
A. Mielke, T. Roubíček, M. Thomas, From damage to delamination in nonlinearly elastic materials at small strains, Journal of Elasticity. The Physical and Mathematical Science of Solids, 109 (2012), pp. 235273.
Abstract
Brittle Griffithtype delamination of compounds is deduced by means of Gammaconvergence from partial, isotropic damage of threespecimensandwichstructures by flattening the middle component to the thickness 0. The models used here allow for nonlinearly elastic materials at small strains and consider the processes to be unidirectional and rateindependent. The limit passage is performed via a double limit: first, we gain a delamination model involving the gradient of the delamination variable, which is essential to overcome the lack of a uniform coercivity arising from the passage from partial damage to delamination. Second, the delaminationgradient is supressed. Noninterpenetration and transmissionconditions along the interface are obtained. 
CH. Heinemann, Ch. Kraus, Existence of weak solutions for CahnHilliard systems coupled with elasticity and damage, Advances in Mathematical Sciences and Applications, 21 (2011), pp. 321359.
Abstract
The CahnHilliard model is a typical phase field approach for describing phase separation and coarsening phenomena in alloys. This model has been generalized to the socalled CahnLarché system by combining it with elasticity to capture nonneglecting 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 CahnHilliard and CahnLarché 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 ratedependent damage processes under certain growth conditions of the energy functional. 
K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the NavierStokesKorteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011), pp. 239254.
Abstract
In this contribution we will study the behaviour of the pressure across phase boundaries in liquidvapour flows. As mathematical model we will consider the static version of the NavierStokesKorteweg 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. 
CH. Kraus, The degenerate and nondegenerate Stefan problem with inhomogeneous and anisotropic GibbsThomson law, European Journal of Applied Mathematics, 22 (2011), pp. 393422.
Abstract
The Stefan problem is coupled with a spatially inhomogeneous and anisotropic GibbsThomson condition at the phase boundary. We show the longtime existence of weak solutions for the nondegenerate Stefan problem with a spatially inhomogeneous and anisotropic GibbsThomson 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 GibbsThomson law in a weak generalized BVformulation. 
A. Mielke, U. Stefanelli, Weighted energydissipation functionals for gradient flows, ESAIM. Control, Optimisation and Calculus of Variations, 17 (2011), pp. 5285.
Abstract
We investigate a globalintime variational approach to abstract evolution by means of the weighted energydissipation 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 timediscretization. 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 reactiondiffusion systems and for energydriftdiffusion systems, Nonlinearity, 24 (2011), pp. 13291346.
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 reactiondiffusion 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. 
H. Garcke, Ch. Kraus, An anisotropic, inhomogeneous, elastically modified GibbsThomson law as singular limit of a diffuse interface model, Advances in Mathematical Sciences and Applications, 20 (2010), pp. 511545.
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 GibbsThomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the EulerLagrange equation of the diffuse phase field energy. 
R. HallerDintelmann, J. Rehberg, Coercivity for elliptic operators and positivity of solutions on Lipschitz domains, Archiv der Mathematik, 95 (2010), pp. 457468.
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 nonzero 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 WaalsCahnHilliard phasefield model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010), pp. 11611186.
Abstract
We study the equilibria of liquidvapor 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 WaalsCahnHilliard 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 twophase 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. 
D. Knees, Ch. Zanini, A. Mielke, Crack growth in polyconvex materials, Physica D. Nonlinear Phenomena, 239 (2010), pp. 14701484.
Abstract
We discuss a model for crack propagation in an elastic body, where the crack path is described apriori. In particular, we develop in the framework of finitestrain elasticity a rateindependent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energyrelease rate is no longer continuous with respect to time and the position of the crack tip. Thus, the model is formulated in terms of the Clarke differential of the energy, generalizing the classical crack evolution models for elasticity with strictly convex energies. We prove the existence of solutions for our model and also the existence of special solutions, where only certain extremal points of the Clarke differential are allowed. 
D. Knees, On global spatial regularity and convergence rates for time dependent elastoplasticity, Mathematical Models & Methods in Applied Sciences, 20 (2010), pp. 18231858.

D. Knees, On global spatial regularity in elastoplasticity with linear hardening, Calculus of Variations and Partial Differential Equations, 36 (2009), pp. 611625.
Abstract
We study the global spatial regularity of solutions of elastoplastic 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);H^{3/2δ}(Ω)) for the displacements and z in L^{∞}((0,T);H^{1/2δ}(Ω)) for the internal variables. The proof is based on a difference quotient technique and a reflection argument. 
S. Heinz, Quasiconvex functions can be approximated by quasiconvex polynomials, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), pp. 795801.

F. Auricchio, A. Mielke, U. Stefanelli, A rateindependent model for the isothermal quasistatic evolution of shapememory materials, Mathematical Methods in the Applied Sciences, 18 (2008), pp. 125164.
Abstract
This note addresses a threedimensional model for isothermal stressinduced transformation in shapememory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rateindependent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasistatic evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics. 
D. Knees, A. Mielke, Energy release rate for cracks in finitestrain elasticity, Mathematical Methods in the Applied Sciences, 31 (2008), pp. 501528.
Abstract
Griffith's fracture criterion describes in a quasistatic setting whether or not a preexisting 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 wellknown 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. 15291569.
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 rateindependent 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 finitestrain elasticity, Mechanics of Advanced Materials and Structures, 15 (2008), pp. 421427.

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; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 187 (2008), pp. 157184.

A. Mielke, M. Ortiz, A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), pp. 494516.

A. Mielke, U. Stefanelli, A discrete variational principle for rateindependent evolution, Advances in Calculus of Variations, 1 (2008), pp. 399431.
Abstract
We develop a globalintime variational approach to the timediscretization of rateindependent processes. In particular, we investigate a discrete version of the variational principle based on the weighted energydissipation functional introduced by A. Mielke and M. Ortiz in ESAIM Control Optim. Calc. Var., 2008. We prove the conditional convergence of timediscrete approximate minimizers to energetic solutions of the timecontinuous problem. Moreover, the convergence result is combined with approximation and relaxation. For a fixed partition the functional is shown to have an asymptotic development by Gamma convergence, cf. G. Anzellotti and S. Baldo (Appl. Math. Optim., 1993), in the limit of vanishing viscosity. 
O. Minet, H. Gajewski, J.A. Griepentrog, J. Beuthan, The analysis of laser light scattering during rheumatoid arthritis by image segmentation, Laser Physics Letters, 4 (2007), pp. 604610.

H. Gajewski, J.A. Griepentrog, A descent method for the free energy of multicomponent systems, Discrete and Continuous Dynamical Systems, 15 (2006), pp. 505528.

D. Knees, Griffithformula and Jintegral for a crack in a powerlaw hardening material, Mathematical Models & Methods in Applied Sciences, 16 (2006), pp. 17231749.

A. Mielke, S. Müller, Lower semicontinuity and existence of minimizers in incremental finitestrain elastoplasticity, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 3 (2006), pp. 233250.

A. Mielke, Necessary and sufficient conditions for polyconvexity of isotropic functions, Journal of Convex Analysis, 12 (2005), pp. 291314.
Contributions to Collected Editions

G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Some remarks on a model for rateindependent damage in thermoviscoelastodynamics, in: MURPHYSHSFS2014: 7th International Workshop on MUltiRate Processes and HYSteresis (MURPHYS) & 2nd International Workshop on Hysteresis and SlowFast 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. 012009/1012009/20.
Abstract
This note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the ratedependent heat equation, and with the momentum balance featuring inertia and viscosity according to KelvinVoigt 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 timedependent Dirichlet loading. 
A. Mielke, Relaxation of a rateindependent phase transformation model for the evolution of microstructure, in: Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure, Workshop, March 1418, 2016, R. Kienzler, D.L. Mcdowell, S. Müller, E.A. Werner, eds., 13 of Oberwolfach Reports, European Mathematical Society, 2016, pp. 840842.

A. Mielke, Multiscale gradient systems and their amplitude equations, in: Dynamics of Pattern, Workshop, Dezember 1622, 2012, 9 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2012, pp. 35883591.

D. Knees, R. Rossi, C. Zanini, A vanishing viscosity approach in damage mechanics, in: Variational Methods for Evolution, Workshop, December 410, 2011, A. Mielke, F. Otto, G. Savaré, U. Stefanelli, eds., 8 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2011, pp. 31533155.

D. Knees, A survey on energy release rates, in: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture, Miniworkshop, April 2429, 2011, 8 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2011, pp. 12161219.

M. Thomas, Modeling and analysis of rateindependent damage and delamination processes, in: Proceedings of the 19th International Conference on Computer Methods in Mechanics (online only), 2011, pp. 16.

A. Mielke, Existence theory for finitestrain 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. 171183.

H. Gajewski, J.A. Griepentrog, A. Mielke, J. Beuthan, U. Zabarylo, O. Minet, Image segmentation for the investigation of scatteredlight images when laseroptically diagnosing rheumatoid arthritis, in: Mathematics  Key Technology for the Future, W. Jäger, H.J. Krebs, eds., Springer, Heidelberg, 2008, pp. 149161.

D. Knees, Energy release rate for cracks in finitestrain elasticity, in: Analysis and Numerics of RateIndependent Processes, Workshop, February 26  March 2, 2007, 4 of Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 2007, pp. 627630.
Preprints, Reports, Technical Reports

M. Hintermüller, M. Holler, K. Papafitsoros, A function space framework for structural total variation regularization with applications in inverse problems, Preprint no. 2437, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2437 .
Abstract, PDF (627 kByte)
In this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semicontinuous envelope (relaxation) of a suitable total variation type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted total variation for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddlepoint problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson loglikelihood data discrepancy terms. Finally, we provide proofofconcept numerical examples where we solve the saddlepoint problem for weighted TV denoising as well as for MR guided PET image reconstruction. 
M. Liero, S. Melchionna, The weighted energydissipation principle and evolutionary Γconvergence for doubly nonlinear problems, Preprint no. 2411, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2411 .
Abstract, PDF (392 kByte)
We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the socalled weighted energydissipation (WED) functional, whose minimizer correspond to solutions of an ellipticintime regularization of the target problems with regularization parameter δ. We investigate the relation between the Γconvergence of the WED functionals and evolutionary Γconvergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γconvergence of functionals or in the sense of evolutionary Γconvergence of functionaldriven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λconvex energy functionals. Finally, we discuss a homogenization problem as an example of application. 
M. Thomas, A comparison of delamination models: Modeling, properties, and applications, Preprint no. 2393, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2393 .
Abstract, PDF (140 kByte)
This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed. 
S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$regularization, Preprint no. 2388, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2388 .
Abstract, PDF (566 kByte)
We address a model for rateindependent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BVregularization. Discrete solutions are obtained using an alternate timediscrete scheme and the VariableADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rateindependent system. Moreover, we present our numerical results for two benchmark problems. 
M. Hintermüller, A. Langer, C.N. Rautenberg, T. Wu, Adaptive regularization for image reconstruction from subsampled data, Preprint no. 2379, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2379 .
Abstract, PDF (1600 kByte)
Choices of regularization parameters are central to variational methods for image restoration. In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in [15], which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach. 
A. Alphonse, Ch.M. Elliott, J. Terra, A coupled ligandreceptor bulksurface 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 reactiondiffusion system of coupled bulksurface equations on a moving domain modelling receptorligand 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 nonmoving 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 Giorgitype arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for timedependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves. 
A. Mielke, M. Mittnenzweig, Convergence to equilibrium in energyreactiondiffusion systems using vectorvalued functional inequalities, Preprint no. 2349, WIAS, Berlin, 2016.
Abstract, PDF (618 kByte)
We discuss how the recently developed energydissipation methods for reactiondi usion systems can be generalized to the nonisothermal 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 logSobolev estimate and variants for lowerorder 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, integrodifferential operators. 
B. Abdellaoui, A. Dieb, E. Valdinoci, A nonlocal concaveconvex 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 concaveconvex nonlinearity and the DirichletNeumann data. 
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 onedimensional 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 reactiondiffusion systems with complexbalanced massaction kinetics, Preprint no. 2326, WIAS, Berlin, 2016.
Abstract, PDF (444 kByte)
We consider reactiondiffusion systems on a bounded domain with noflux boundary conditions. All reactions are given by the massaction law and are assumed to satisfy the complexbalance 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 energydissipation estimates, which guarantee exponential decay of the relative entropy, all of which rely on the logSobolev estimate and suitable handling of the reaction terms as well as the massconservation 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 selfdriven 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 blowup limits are minimizers of the AltCaffarelliFriedman functional. Namely, the energy of the problem somehow linearizes in the blowup limit. We also develop a detailed optimal regularity theory for the minimizers and for their free boundaries. 
L. Avena, O. Gün, M. Hesse, The parabolic Anderson model on the hypercube, Preprint no. 2319, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2319 .
Abstract, PDF (240 kByte)
We consider the parabolic Anderson model (PAM) on the ndimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the kth largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and kth largest potential. One of our main motivations in this article is to investigate the mutationselection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutationselection model as follows: a population initially concentrated at the site of the kth best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape. 
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 adhoc 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. 
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 scaloric, up to a small error. The case of nonelliptic and nonparabolic operators is taken into account as well. 
S. Patrizi, E. Valdinoci, Longtime behavior for crystal dislocation dynamics, Preprint no. 2302, WIAS, Berlin, 2016.
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).
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. 
M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, Analytical aspects of spatially adapted total variation regularisation, Preprint no. 2293, WIAS, Berlin, 2016.
Abstract, PDF (877 kByte)
In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is wellposed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions. 
F. Flegel, Localization of the principal Dirichlet eigenvector in the heavytailed random conductance model, Preprint no. 2290, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2290 .
Abstract, PDF (567 kByte)
We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^q]<∞ <¼, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γrm c = ¼ is sharp. Indeed, other recent results imply that for γ>¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, BorelCantelli arguments, the RayleighRitz formula, results from percolation theory, and path arguments. 
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 R^{n} 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. 
R.I.A. Patterson, D.R.M. Renger, Dynamical large deviations of countable reaction networks under a weak reversibility condition, Preprint no. 2273, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2273 .
Abstract, PDF (343 kByte)
A dynamic large deviations principle for a countable reaction network including coagulationfragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space. 
M. Cozzi, S. Dipierro, E. Valdinoci, Planelike interfaces in longrange 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 longrange 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 longrange Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for longrange 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 longrange 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 doublewell function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as Gammaconvergence, 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. 
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 OhtaKawasaki 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 volumeconstrained 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 onedimension, 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.

M. Hintermüller, C.N. Rautenberg, T. Wu, A. Langer, Optimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests, Preprint no. 2236, WIAS, Berlin, 2016.
Abstract, PDF (6570 kByte)
Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on. 
M. Hintermüller, C.N. Rautenberg, Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory, Preprint no. 2235, WIAS, Berlin, 2016.
Abstract, PDF (417 kByte)
A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchelpredual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lowerlevel problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upperlevel objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes. 
J. Haskovec, S. Hittmeir, P. Markowich, A. Mielke, Decay to equilibrium for energyreactiondiffusion systems, Preprint no. 2233, WIAS, Berlin, 2016.
Abstract, PDF (436 kByte)
We derive thermodynamically consistent models of reactiondiffusion 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 diffusionreaction bipolar energy transport system, and a driftdiffusionreaction energy transport system with confining potential. We prove corresponding entropyentropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L^{1} using CziszarKullbackPinsker type inequalities. 
K. Disser, M. Liero, J. Zinsl, On the evolutionary Gammaconvergence 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 massaction 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 Econvergence result via Γconvergence 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 pseudometric. 
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 twodimensional 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. 
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 (590 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 stateshape 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 timediscretised 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. Liero, A. Mielke, G. Savaré, Optimal entropytransport problems and a new HellingerKantorovich distance between positive measures, Preprint no. 2207, WIAS, Berlin, 2016.
Abstract, PDF (1154 kByte)
We develop a full theory for the new class of Optimal EntropyTransport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic EntropyTransport problems and introduce the new HellingerKantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the wellknown HellingerKakutani and KantorovichWasserstein distances. 
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. 
A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Nonequilibrium thermodynamical principles for chemical reactions with massaction 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 nonlinear relation between thermodynamic fluxes and free energy driving force. 
S. Dipierro, N. Soave, E. Valdinoci, On stable solutions of boundary reactiondiffusion equations and applications to nonlocal problems with Neumann data, Preprint no. 2152, WIAS, Berlin, 2015.
Abstract, PDF (355 kByte)
We study reactiondiffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar´etype 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. 
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 phasefield system, Preprint no. 2133, WIAS, Berlin, 2015.
Abstract, PDF (295 kByte)
In the present contribution the sliding mode control (SMC) problem for a phasefield model of Caginalp type is considered. First we prove the wellposedness and some regularity results for the phasefield 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 nonlocal 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. 
M. Liero, S. Reichelt, Homogenization of CahnHilliardtype equations via evolutionary Gammaconvergence, Preprint no. 2114, WIAS, Berlin, 2015.
Abstract, PDF (455 kByte)
In this paper we discuss two approaches to evolutionary Γconvergence 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 Γconvergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the timedependent 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 energydissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for CahnHilliardtype equations. Using the method of weak and strong twoscale convergence via periodic unfolding, we show that the energy and dissipation functionals Γconverge. In conclusion, we will give specific examples for the applicability of each of the two approaches. 
M. Cozzi, E. Valdinoci, Planelike minimizers for a nonlocal GinzburgLandautype energy in a periodic medium, Preprint no. 2108, WIAS, Berlin, 2015.
Abstract, PDF (1768 kByte)
We consider a nonlocal 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 nonlocal 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 timedependent. 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 FisherKPP 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 reactiondiffusion equation of FisherKPP type inside an infinite cylindrical domain in $R^N+1$, coupled with a reactiondiffusion 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 stripshaped field bounded by them. 
S. Dipierro, E. Valdinoci, Continuity and density results for a onephase nonlocal free boundary problem, Preprint no. 2101, WIAS, Berlin, 2015.
Abstract, PDF (358 kByte)
We consider a onephase 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 socalled 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 SobolevBochner 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. 
X. RosOton, J. Serra, E. Valdinoci, Pohozaev identities for anisotropic integrodifferential operators, Preprint no. 2080, WIAS, Berlin, 2015.
Abstract, PDF (339 kByte)
We establish Pohozaev identities and integration by parts type formulas for anisotropic integrodifferential 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

A. Alphonse, A coupled bulksurface reactiondiffusion system on a moving domain, Workshop ``Emerging Developments in Interfaces and Free Boundaries'', January 23  28, 2017, Mathematisches Forschungszentrum Oberwolfach, January 25, 2017.

A. Alphonse, Optimal control of elliptic and parabolic quasivariational inequalities, Annual Meeting of the DFG Priority Programme 1962, October 9  11, 2017, Kremmen (Sommerfeld), October 10, 2017.

M. Liero, On entropytransport problems and the HellingerKantorovich distance, Seminar of Team EDPAIRSEACVGI, Université Grenoble Alpes, Laboratoire Jean Kuntzmann, Grenoble, France, January 26, 2017.

S.M. Stengl, Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms and risk aversion (with Deborah Gahururu), Annual Meeting of the DFG Priority Programme 1962, October 9  11, 2017, Kremmen (Sommerfeld), October 9, 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.

M. Thomas, Rateindependent delamination processes in viscoelasticity, Miniworkshop on Dislocations, Plasticity, and Fracture, February 13  16, 2017, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, February 15, 2017.

M. Hintermüller, Bilevel optimization and applications in imaging, Workshop ``Emerging Developments in Interfaces and Free Boundaries'', January 22  28, 2017, Mathematisches Forschungsinstitut Oberwolfach.

M. Hintermüller, Bilevel optimization and applications in imaging, Mathematisches Kolloquium, Universität Wien, Austria, January 18, 2017.

M. Hintermüller, Bilevel optimization and some ``parameter learning'' applications in image processing, LMS Workshop ``Variational Methods Meet Machine Learning'', September 18, 2017, University of Cambridge, Centre for Mathematical Sciences, UK, September 18, 2017.

M. Hintermüller, Nonsmooth structures in PDEconstrained optimization, Mathematisches Kolloquium, Universität DuisburgEssen, Fakultät für Mathematik, Essen, January 11, 2017.

M. Hintermüller, On (pre)dualization, dense embeddings of convex sets, and applications in image processing, Isaac Newton Institute, Cambridge, UK, August 30, 2017.

M. Hintermüller, On (pre)dualization, dense embeddings of convex sets, and applications in image processing, University College London, Centre for Inverse Problems, UK, October 27, 2017.

M. Hintermüller, Recent trends in PDEconstrained optimization with nonsmooth structures, Fourth Conference on Numerical Analysis and Optimization (NAOIV2017), January 2  5, 2017, Sultan Qaboos University, Muscat, Oman, January 4, 2017.

M. Liero, The HellingerKantorovich distance as natural generalization of optimal transport distance to (scalar) reactiondiffusion equations, Variational Methods for Evolution, November 12  17, 2017, Mathematisches Forschungszentrum Oberwolfach, November 14, 2017.

A. Mielke, A geometric approach to reactiondiffusion equations, Institutskolloquium, Universität Potsdam, Institut für Mathematik, January 25, 2017.

A. Mielke, Entropyinduced geometry for classical and quantum Markov semigroups, SMS Colloquium, University College Cork, School of Mathematical Science, Ireland, September 11, 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 energyreactiondiffusion systems, Analysis Seminar, University of Pavia, Department of Mathematics, Italy, March 21, 2017.

M. Mittnenzweig, Variational methods for quantum master equations, Barcelona Graduate School of Mathematics Junior Meeting, October 9  10, 2017, Berlin Mathematical School, Spain, October 10, 2017.

E. Cinti, Quantitative flatness results and BV estimates for nonlocal minimal surfaces, Workshop ``Calculus of Variations'', July 11  15, 2016, Mathematisches Forschungsinstitut Oberwolfach, July 12, 2016.

E. Cinti, Quantitative flatness results and BVestimates for nonlocal minimal surfaces, BruxellesTorino talks in PDE's, May 2  5, 2016, Università degli Studi di Torino, Dipartimento di Matematica ``Giuseppe Peano'', Italy, May 3, 2016.

E. Cinti, Quantitative flatness results and BVestimates for nonlocal minimal surfaces, 9th European Conference on Elliptic and Parabolic Problems, May 23  27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 23, 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, Econvergence to the quasisteadystate 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, Homogenization of CahnHilliardtype 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 CahnHilliardtype equations via evolutionary Gammaconvergence, Workshop ``Patterns of Dynamics'', Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 25  29, 2016.

S. Reichelt, Homogenization of CahnHilliardtype 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, EDPconvergence 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 Gammaconvergence 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 Gammaconvergence 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 Gammaconvergence, 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 AllenCahn 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 reactiondiffusion systems and optimal entropytransport 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, On EntropyTransport problems and distances between positive measures, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22  26, 2016, WIAS Berlin, Berlin, February 25, 2016.

M. Liero, On entropytransport problems and the HellingerKantorovich distance, Followup Workshop to Junior Hausdorff Trimester Program ``Optimal Transportation'', August 29  September 2, 2016, Hausdorff Research Institute for Mathematics, Bonn, August 30, 2016.

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

D.R.M. Renger, Functions of bounded variation with an infinitedimensional codomain, Meeting in Applied Mathematics and Calculus of Variations, September 13  16, 2016, Università di Roma ``La Sapienza'', Dipartimento di Matematica ``Guido Castelnuovo'', Italy, September 16, 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, Capillarity problems with nonlocal surface tension energies, Columbia Geometry & Analysis Seminar, Columbia University in the City of New York, Department of Mathematics, USA, September 16, 2016.

E. Valdinoci, Interior and boundary properties of nonlocal minimal surfaces, Calcul des Variations & EDP, Université AixMarseille, 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 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 surface, JustusLiebigUniversität Gießen, Fakultät für Mathematik, February 10, 2016.

E. Valdinoci, Nonlocal minimal surfaces, a geometric and analytic insight, Seminar on Differential Geometry and Analysis, OttovonGuerickeUniversität 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.

T. Wu, Bilevel optimization and applications in imaging sciences, August 24  25, 2016, Shanghai Jiao Tong University, Institute of Natural Sciences, China.

M. Thomas, Coupling rateindependent and ratedependent processes: Delamination models in viscoelastodynamics, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, June 10, 2016.

M. Thomas, Coupling rateindependent and ratedependent 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 rateindependent and ratedependent processes: Damage & delamination in viscoelastodynamics, 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 viscoelastodynamics, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, special session ``Ratedependent and Rateindependent 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 viscoelastodynamics, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22  26, 2016, WIAS Berlin, Berlin, February 26, 2016.

M. Thomas, Nonsmooth 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, Rateindependent evolution of sets, INdAMISIMM 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, Rateindependent evolution of sets & application to fracture processes, Seminar on Analysis, Kanazawa University, Institute of Science and Engineering, Kanazawa, Japan, October 28, 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, K. Papafitsoros, C. Rautenberg, A fine scale analysis of spatially adapted total variation regularisation, Imaging, Vision and Learning based on Optimization and PDEs, Bergen, Norway, August 29  September 1, 2016.

M. Hintermüller, Adaptive finite elements in total variation based image denoising, SIAM Conference on Imaging Science, Minisymposium ``Leveraging Ideas from Imaging Science in PDEconstrained Optimization'', May 23  26, 2016, Albuquerque, USA, May 24, 2016.

M. Hintermüller, Bilevel optimization and applications in imaging, Imaging, Vision and Learning based on Optimization and PDEs, August 29  September 1, 2016, Bergen, Norway, August 30, 2016.

M. Hintermüller, Bilevel optimization for a generalized totalvariation model, SIAM Conference on Imaging Science, Minisymposium ``NonConvex Regularization Methods in Image Restoration'', May 23  26, 2016, Albuquerque, USA, May 26, 2016.

M. Hintermüller, Optimal selection of the regularisation function in a localised TV model, SIAM Conference on Imaging Science, Minisymposium ``Analysis and Parameterisation of Derivative Based Regularisation'', May 23  26, 2016, Albuquerque, USA, May 24, 2016.

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

M. Hintermüller, Shape and topological sensitivities in mathematical image processing, BMS Summer School ``Mathematical and Numerical Methods in Image Processing'', July 25  August 5, 2016, Berlin Mathematical School, Technische Universität Berlin, HumboldtUniversität zu Berlin, Berlin, August 4, 2016.

M. Hintermüller, Towards sharp stationarity conditions for classes of optimal control problems for variational inequalities of the second kind, International INdAM Conference ``Optimal Control for Evolutionary PDEs and Related Topics (OCERTO 2016)'', June 20  24, 2016, Cortona, Italy, June 20, 2016.

A. Mielke, Entropyentropy production estimates for energyreaction 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, Evolutionary Gammaconvergence, 2nd CENTRAL School on Analysis and Numerics for Partial Differential Equations, August 29  September 2, 2016, HumboldtUniversität zu Berlin, Institut für Mathematik.

A. Mielke, Evolutionary relaxation for a rateindependent phasetransformation model, Workshop ``Mechanics of Materials: Mechanics of Interfaces and Evolving Microstructure'', March 14  18, 2016, Mathematisches Forschungszentrum Oberwolfach, March 14, 2016.

A. Mielke, Global existence for finitestrain viscoplasticity via the energydissipation 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 reactiondiffusion equation, Mathematisches Kolloquium, Westfälische WilhelmsUniversität, Institut für Mathematik, Münster, April 28, 2016.

A. Mielke, Microstructure evolution via relaxation for a rateindependent elastic twophase model, Joint Annual Meeting of DMV and GAMM, Session ``Applied Analysis'', March 7  11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

A. Mielke, On a model for the evolution of microstructures in solids  Evolutionary relaxation, KTGUIMU Mathematics Colloquia, March 30  31, 2016, Kyoto University, Department of Mathematics, Japan, March 31, 2016.

A. Mielke, On entropic gradient structures for classical and quantum Markov processes with detailed balance, Pure Analysis and PDEs Seminar, Imperial College London, Department of Mathematics, UK, May 11, 2016.

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

A. Mielke, On the geometry of reaction and diffusion, INdAMISIMM 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 reactiondiffusion equations, Pure Analysis and PDEs Seminar, Imperial College London, Department of Mathematics, UK, May 12, 2016.

A. Mielke, Rateindependent microstructure evolution via relaxation of a twophase model, Workshop ``Advances in the Mathematical Analysis of Material Defects in Elastic Solids'', June 6  10, 2016, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, June 10, 2016.

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 isoperimetric inequality via the ABP method, Università di Bologna, Dipartimento di Matematica, Bologna, Italy, July 17, 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 CahnHilliard 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 CahnHilliard/NavierStokes systems with nonlocal interactions, Control Theory and Related Topics, April 13  14, 2015, Politecnico di Milano, Italy, April 13, 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, On dissipation distances for reactiondiffusion equations  The HellingerKantorovich 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.R.M. Renger, The inverse problem: From gradient flows to large deviations, Workshop ``Analytic Approaches to Scaling Limits for Random System'', January 26  30, 2015, Universität Bonn, Hausdorff Research Institute for Mathematics, January 26, 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?, PerlenKolloquium, Universität Basel, Fachbereich Mathematik, Switzerland, May 22, 2015.

F. Flegel, Localization of the first Dirichleteigenvector in the heavytailed random conductance model, Summer School 2015 of the RTG 1845 BerlinPotsdam ``Stochastic Analysis with Applications in Biology, Finance and Physics'', September 28  October 3, 2015, Levico Terme, Italy, October 1, 2015.

F. Flegel, Localization of the first Dirichleteigenvector in the heavytailed random conductance model, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26  August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, July 30, 2015.

M. Thomas, Analysis of nonsmooth PDE systems with application to material failuretowards 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 rateindependent and ratedependent processes: Existence results, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Pavia, Italy, March 5, 2015.

M. Thomas, Coupling rateindependent and ratedependent processes: Evolutionary Gammaconvergence 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 rateindependent and ratedependent 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 rateindependent and ratedependent 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, From adhesive contact to brittle delamination in viscoelastodynamics, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 2, 2015.

M. Thomas, From adhesive contact to brittle delamination in viscoelastodynamics, Workshop on CENTRAL Trends in PDEs, November 12  13, 2015, University of Vienna, Faculty of Mathematics, Vienna, Austria, November 13, 2015.

M. Thomas, Rateindependent damage models with spatial BVregularization  Existence & fine properties of solutions, Oberseminar ``Angewandte Analysis'', Universität Freiburg, Abteilung für Angewandte Mathematik, Freiburg, February 10, 2015.

A. Mielke, A mathematical approach to finitestrain 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 rateindependent 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, EDPconvergence and the limit from diffusion to reaction, 3rd Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 30  October 2, 2015, Universität Kassel, Institut für Mathematik, Kassel, October 2, 2015.

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

A. Mielke, Evolutionary $Gamma$convergence for gradient systems explained via applications, Symposium ``Variational Methods for Stationary and Evolutionary Problems'', University of Warwick, Mathematics Institute, Warwick, UK, May 12, 2015.

A. Mielke, Existence results in finitestrain 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, Mathematical modeling for finitestrain 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 multiplicative strain decomposition in finitestrain 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.

C. Kreisbeck, Thinfilm limits of functionals on Afree vector fields and applications, Workshop on Trends in NonLinear Analysis 2014, July 31  August 1, 2014, Instituto Superior Técnico, Departamento de Matemática, Lisbon, Portugal, August 1, 2014.

C. Kreisbeck, Thinfilm limits of functionals on Afree 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, Thinfilm limits of functionals on Afree vector fields and applications, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, July 16, 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 phasetransformation model, 13th GAMM Seminar on Microstructures, January 17  18, 2014, RuhrUniversität Bochum, Lehrstuhl für Mechanik  Materialtheorie, January 18, 2014.

M. Liero, On dissipation distances for reactiondiffusion equations  The HellingerKantorovich 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 reactiondiffusion equations  The HellingerKantorovich distance, RIPE60  Rate Independent Processes and Evolution Workshop, June 24  26, 2014, Prague, Czech Republic, June 24, 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 Nonlinéaires, 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, HeriotWatt 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 Nonlocal and Nonlinear 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, IEMathGranada, 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.

D. Knees, A quasilinear differential inclusion for viscous and rateindependent damage systems in nonsmooth domains, Analysis & Stochastics Seminar, Technische Universität Dresden, Institut für Analysis, January 16, 2014.

M. Thomas, A stressdriven localsolution 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 rateindependent processes in viscoelastic materials, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, March 18, 2014.

M. Thomas, Existence and stability results for rateindependent 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, FriedrichAlexander Universität ErlangenNürnberg, March 10  14, 2014.

M. Thomas, Rateindependent systems with viscosity and inertia: Existence and evolutionary Gammaconvergence, Workshop ``Variational Methods for Evolution'', December 14  20, 2014, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic materials, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 8, 2014.

M. Thomas, Rateindependent, partial damage in thermoviscoelastic 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, Rateindependent, partial damage in thermoviscoelastic materials with inertia, Oberseminar ``Analysis und Angewandte Mathematik'', Universität Kassel, Institut für Mathematik, December 1, 2014.

M. Thomas, Stressdriven localsolution 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, FriedrichAlexander Universität ErlangenNürnberg, March 11, 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, FriedrichAlexander Universität ErlangenNürnberg, March 13, 2014.

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

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

A. Mielke, Modeling jumps in rateindependent systems using balancedviscosity solutions, 7th International Workshop on MultiRate Processes & Hysteresis, 2nd International Workshop on Hysteresis and SlowFast Systems (MURPHYSHSFS2014), April 7  11, 2014, WIAS Berlin, April 8, 2014.

A. Mielke, On gradient structures and dissipation distances for reactiondiffusion systems, Kolloquium ``Angewandte Mathematik'', FriedrichAlexanderUniversität ErlangenNürnberg, Department Mathematik, July 3, 2014.

A. Mielke, On gradient structures for reactiondiffusion systems, Joint Analysis Seminar, RheinischWestfälische Technische Hochschule Aachen (RWTH), Institut für Mathematik, February 4, 2014.

A. Mielke, A reactiondiffusion equation as a HellingerKantorovich 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 BritishGerman Frontiers of Science Symposium, Potsdam, March 6  9, 2014.

S. Neukamm, Characterization and approximation of macroscopic properties with homogenization, 4th BritishGerman Frontiers of Science Symposium, March 6  9, 2014, Alexander von HumboldtStiftung, Potsdam, March 7, 2014.

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

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

H. Stephan, Inequalities for Markov operators and applications to forward and backward PDEs, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 88: Stochastic Processes and Spectral Theory for Partial Differential Equations and Boundary Value Problems, July 7  11, 2014, Madrid, Spain, July 8, 2014.

K. Disser, Entropic gradient structures for reversible Markov chains and the passage to Wasserstein FokkerPlanck, 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 FokkerPlanck equation, Oberseminar Analysis, MartinLutherUniversität HalleWittenberg, Institut für Mathematik, Halle, November 20, 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 CahnHilliard 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 CahnHilliard systems coupled with complete damage processes, DIMO2013  Diffuse Interface Models, Levico Terme, Italy, September 10  13, 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 reactiondiffusion 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 for driftreactiondiffusion 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.

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

D. Knees, A vanishing viscosity approach to a rateindependent 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.

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 NavierStokesAllenCahn 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 thermoviscoelastic 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 rateindependent 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, Local versus energetic solutions in rateindependent brittle delamination, DIMO2013  Diffuse Interface Models, September 10  13, 2013, Levico Terme, Italy, September 13, 2013.

M. Thomas, A stressdriven 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 stressdriven 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 rateindependent 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 microstructure, Oberseminar zur Analysis, Universität DuisburgEssen, Fachbereich Mathematik, Essen, January 29, 2013.

H. Hanke, Derivation of an effective damage model with nonperiodic evolving microstructure, 12th GAMM Seminar on Microstructures, February 8  9, 2013, HumboldtUniversität zu Berlin, Institut für Mathematik, February 9, 2013.

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

A. Mielke, Introduction to evolutionary Gamma convergence for gradient systems, School ``Multiscale and Multifield Representations of Condensed Matter Behavior'', November 25  29, 2013, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

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

P. Gussmann, Linearisierte Elastizität als Grenzwert finiter Elastizität im Falle von Schlitzgebieten, Jahrestagung der Deutsche MathematikerVereinigung (DMV), Studierendenkonferenz, September 17  20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 20, 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 ratedependent complete damage processes, Materialmodellierungsseminar, WIAS, Berlin, October 31, 2012.

CH. Heinemann, Kopplung von Phasenseparation und Schädigung in elastischen Materialien, LeibnizDoktorandenForum der Sektion D, Berlin, June 7  8, 2012.

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

D. Knees, Global spatial regularity for elastic fields with cracks and contract, 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.

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

CH. Kraus, CahnLarché 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 GibbsThomson law, 6th European Congress of Mathematics, July 2  6, 2012, Cracow, Poland, July 5, 2012.

M. Thomas, A model for rateindependent, brittle delamination in thermoviscoelasticity, 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 29, 2012.

M. Thomas, A model for rateindependent, brittle delamination in thermoviscoelasticity, INDAM Workshop PDEs for Multiphase Advanced Materials (ADMAT2012), September 17  21, 2012, Cortona, Italy, September 17, 2012.

M. Thomas, Analytical aspects of rateindependent damage models with spatial BVregularization, Seminar, SISSA  International School for Advanced Studies, Functional Analysis and Applications, Trieste, Italy, November 28, 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, Rateindependent evolution of sets, Variational Models and Methods for Evolution, Levico, Italy, September 10  12, 2012.

H. Hanke, Derivation of an effective damage evolution model using twoscale 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, Finitestrain 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, From smallstrain to finitestrain elastoplasticity via evolutionary Gamma convergence, Variational Models and Methods for Evolution, September 10  12, 2012, Centro Internazionale per la Ricerca Matematica (CIRM) and Istituto di Matematica Applicata e Tecnologie Informatiche/Consiglio Nazionale delle Ricerche (IMATICNR), Levico, Italy, September 11, 2012.

A. Mielke, Multiscale gradient systems and their amplitude equations, Workshop ``Dynamics of Patterns'', December 17  21, 2012, Mathematisches Forschungsinstitut Oberwolfach, December 18, 2012.

A. Mielke, On gradient flows and reactiondiffusion systems, Institutskolloquium, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, December 3, 2012.

A. Mielke, Smallstrain elastoplasticity is the evolutionary Gamma limit of finitestrain elastoplasticity, International Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2012), September 3  6, 2012, Israel Institute of Technology (Technion), Faculty of Aerospace Engineering, Haifa, September 4, 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.

S. Jansen, Large deviations for interacting manyparticle systems in the Saha regime, BerlinLeipzig Seminar on Analysis and Probability Theory, July 8, 2011, Technische Universität Clausthal, Institut für Mathematik, July 8, 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.

S. Heinz, Regularizations and relaxations of timecontinuous 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.

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

D. Knees, A survey on energy release rates, MiniWorkshop ``Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture'', April 24  29, 2011, Mathematisches Forschungsinstitut Oberwolfach, April 26, 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.

D. Knees, A vanishing viscosity approach in fracture mechanics, Seminar on Partial Differential Equations, Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague, March 1, 2011.

D. Knees, Numerical convergence analysis for a vanishing viscosity model in fracture mechanics, 10th GAMM Seminar on Microstructures, January 20  22, 2011, Technische Universität Darmstadt, Fachbereich Mathematik, January 21, 2011.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, BerlinLeipzig Seminar on Analysis and Probability Theory, Technische Universität Clausthal, Institut für Mathematik, July 8, 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, Multiscale problems in systems driven by functionals, ISAMTopMath Summer School 2011 on Variational Methods, September 12  16, 2011, Technische Universität München, Fakultät für Mathematik.

M. Thomas, From damage to delamination, 82th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2011), Session on Damage and Fracture Mechanics, April 18  21, 2011, Technische Universität Graz, Austria, April 21, 2011.

M. Thomas, Modeling and analysis of rateindependent 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.

M. Thomas, Delamination in viscoelastic materials with thermal effects, Oberseminar ``Mathematik in den Naturwissenschaften'', Universität Würzburg, Institut für Mathematik, November 24, 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.

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.

CH. Kraus, An inhomogeneous, anisotropic and elastically modified GibbsThomson 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 phasefield quantities in the sharp interface limit, 6th Singular Days 2010, April 29  May 1, 2010, WIAS, Berlin, April 30, 2010.

A. Mielke, A mathematical model for the evolution of microstructures in elastoplasticity, Fifth International Conference on Multiscale Materials Modeling, Symposium on Mathematical Methods, October 4  8, 2010, Fraunhofer Institut für Werkstoffmechanik (IWM), Freiburg, October 4, 2010.

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

A. Mielke, Gradient structures for reactiondiffusion systems and semiconductor equations, 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 24, 2010.

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

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, NWFI 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 spacetime discretizations of rateindependent processes, 8th GAMM Seminar on Microstructures, January 15  17, 2009, Universität Regensburg, NWFI Mathematik, January 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.

H.Chr. Kaiser, Transient KohnSham theory, Jubiläumssymposium ``Licht  Materialien  Modelle'' (100 Jahre Innovation aus Adlershof), BerlinAdlershof, September 7  8, 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.

A. Petrov, Some mathematical results for a model of thermallyinduced phase transformations in shapememory materials, sc MatheonICM Workshop on Free Boundaries and Materials Modeling, March 17  18, 2008, WIAS, March 18, 2008.

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

CH. Kraus, Ein Phasenfeldmodell vom CahnHilliardTyp im singulären Grenzwert, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, April 25, 2008.

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

A. Petrov, On the convergence for kinetic variational inequality to quasistatic variational inequality with application to elasticplastic 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 shapememory alloys, Workshop ``Analysis and Numerics of RateIndependent Processes'', February 26  March 2, 2007, Mathematisches Forschungsinstitut Oberwolfach, February 27, 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.

A. Petrov, Mathematical result on the stability of elasticplastic 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 liquidvapor system in the sharp interface limit, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, July 18, 2006.

CH. Kraus, Equilibria conditions in the sharp interface limit of the van der WaalsCahnHilliard 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 WaalsCahnHilliard model, PolishGerman Workshop ``Modeling Structure Formation'', Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Poland, September 8, 2006.

CH. Kraus, On the sharp limit of the Van der WaalsCahnHilliard 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 WaalsCahnHilliard model, Workshop ``MicroMacro Modeling and Simulation of LiquidVapor 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.

H.Chr. Kaiser, Modeling and quasi3D simulation of indium grains in (In,Ga)N/GaN quantum wells by means of density functional theory, Physikalisches Kolloquium, Brandenburgische Technische Universität, Lehrstuhl Theoretische Physik, Cottbus, February 15, 2005.

H.Chr. Kaiser, Quasi3D simulation of multiexcitons by means of density functional theory, Oberseminar ``Numerik/Wissenschaftliches Rechnen'', MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, January 11, 2005.

H.Chr. Kaiser, Spectral resolution of a velocity field on the boundary of a Lipschitz domain, 2nd Joint Meeting of AMS, DMV, ÖMG, June 16  19, 2005, Johannes GutenbergUniversität, Mainz, June 16, 2005.

H.Chr. Kaiser, Density functional theory for multiexcitons in quantum boxes, ``Molecular Simulation: Algorithmic and Mathematical Aspects'', Institut Henri Poincaré, Paris, France, December 1  3, 2004.
External Preprints

D.A. Gomes, S. Patrizi, Obstacle meanfield game problem, Preprint no. arXiv:1410.6942, Cornell University Library, arXiv.org, 2014.
Abstract
In this paper, we introduce and study a firstorder meanfield game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and powerlike nonlinearities. Since the obstacle operator is not differentiable, the equations for firstorder 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 twoscale expansion in stochastic homogenization of discrete elliptic equations, Preprint no. 41, MaxPlanckInstitut 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 twoscale asymptotic expansion has the same scaling as in the periodic case. In particular the L^{2}norm in probability of the H^{1}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. 
D. Knees, Griffithformula and Jintegral for a crack in a powerlaw hardening material, Preprint no. 2005/12, Universität Stuttgart, SFB 404, 2005.
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Applications
 Diffusion models in statistical physics
 Modeling of phase separation and damage in modern materials
 Modeling of thin films and nano structures on substrates
 Modeling, Simulation and Optimization for Biomedical Applications
 Nonlinear material models, multifunctional materials and hysteresis in connection with elastoplastic processes
 Particlebased modeling in the Sciences
 Quantum models for semiconductors