Free boundary problems for partial differential equation describe problems such that a partial differential equation is considered on a domain depending on the solution to the equation.

Properties of solutions to free boundary problems:
Existence, uniqueness, regularity, growth behavior

Phase-field approximation of free boundary problems:
Regularized versions of the free boundary problems
Compatibility with the 2nd law of thermodynamics

Nonlocal free boundary problems:
Due to boundary effects or long-range particle interactions
Nonlocal equations in models of biological environments:
Fractional quantum mechanics

Free boundary problems arising in models for phase transitions:
Free boundary problems in energy technology:

Free-boundary problems in the thin-film geometry
Complex liquids such as polymer mixtures or suspensions
Highly anisotropic solid thin films

Analysis of crystal dislocations:
Evolution of the atom dislocation function
Deducing the macroscopic properties from the microscopic behavior of the system

 


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

  • 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).

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

  • P. Colli, N. Kenmochi, J. Sprekels, eds., Dissipative Phase Transitions, 71 of Series on Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 2006, xii+300 pages, (Collection Published).

  Articles in Refereed Journals

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017) pp. 2518--2546.
    Abstract
    In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in citeHZO. The model consists of a Cahn-Hilliard equation for the tumor cell fraction $vp$ coupled to a reaction-diffusion equation for a function $s$ representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

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

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

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

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

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

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

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

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

  • M. Korzec, A. Münch, E. Süli, B. Wagner, Anisotropy in wavelet based phase field models, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 21 (2016) pp. 1167--1187.
    Abstract
    Anisotropy is an essential feature of phase-field models, in particular when describing the evolution of microstructures in solids. The symmetries of the crystalline phases are reflected in the interfacial energy by introducing corresponding directional dependencies in the gradient energy coefficients, which multiply the highest order derivative in the phase-field model. This paper instead considers an alternative approach, where the anisotropic gradient energy terms are replaced by a wavelet analogue that is intrinsically anisotropic and linear. In our studies we focus on the classical coupled temperature - Ginzburg-Landau type phase-field model for dendritic growth. For the resulting derivative-free wavelet analogue existence, uniqueness and continuous dependence on initial data for weak solutions is proved. The ability to capture dendritic growth similar to the results obtained from classical models is investigated numerically.

  • W. Dreyer, C. Guhlke, R. Müller, A new perspective on the electron transfer: Recovering the Butler--Volmer equation in non-equilibrium thermodynamics, Physical Chemistry Chemical Physics, 18 (2016) pp. 24966--24983.
    Abstract
    Understanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating.

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

  • M. Hintermüller, S. Rösel, A duality-based path-following semismooth Newton method for elasto-plastic contact problems, Journal of Computational and Applied Mathematics, 292 (2016) pp. 150--173.

  • M. Hintermüller, Th. Surowiec, A bundle-free implicit programming approach for a class of elliptic MPECs in function space, Mathematical Programming Series A, 160 (2016) pp. 271--305.

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

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

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

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

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

  • D. Peschka, Thin-film free boundary problems for partial wetting, Journal of Computational Physics, 295 (2015) pp. 770--778.
    Abstract
    We present a novel framework to solve thin-film equations with an explicit non-zero 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 thin-film equations coupled to an arbitrary Lagrangian-Eulerian 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.

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

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

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

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

  • L. Caffarelli, O. Savin , E. Valdinoci, Minimization of a fractional perimeter-Dirichlet integral functional, Annales de l'Institut Henri Poincare. Analyse Non Lineaire, 32 (2015) pp. 901--924.

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

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

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

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

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

  • M.G. Hennessy, V.M. Burlakov, A. Münch, B. Wagner, A. Goriely, Controlled topological transitions in thin-film phase separation, SIAM Journal on Applied Mathematics, 75 (2015) pp. 38--60.
    Abstract
    In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

  • R. Huth, S. Jachalski, G. Kitavtsev, D. Peschka, Gradient flow perspective on thin-film bilayer flows, Journal of Engineering Mathematics, 94 (2015) pp. 43--61.
    Abstract
    We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using Gamma-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.

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

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

  • G. Aki, W. Dreyer, J. Giesselmann, Ch. Kraus, A quasi-incompressible diffuse interface model with phase transition, Mathematical Models & Methods in Applied Sciences, 24 (2014) pp. 827--861.
    Abstract
    This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier-Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier-Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

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

  • L. Blank, M.H. Farshbaf Shaker, H. Garcke, V. Styles, Relating phase field and sharp interface approaches to structural topology optimization, ESAIM. Control, Optimisation and Calculus of Variations, 20 (2014) pp. 1025--1058.
    Abstract
    A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement.

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

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

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

  • C. Bertoglio, A. Caiazzo, A tangential regularization method for backflow stabilization in hemodynamics, Journal of Computational Physics, 261 (2014) pp. 162--171.
    Abstract
    In computational simulations of fluid flows, instabilities at the Neumann boundaries may appear during backflow regime. It is widely accepted that this is due to the incoming energy at the boundary, coming from the convection term, which cannot be controlled when the velocity field is unknown. We propose a stabilized formulation based on a local regularization of the fluid velocity along the tangential directions on the Neumann boundaries. The stabilization term is proportional to the amount of backflow, and does not require any further assumption on the velocity profile. The perfomance of the method is assessed on a two- and three-dimensional Womersley flows, as well as considering a hemodynamic physiological regime in a patient-specific aortic geometry.

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

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

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

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

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

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

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

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

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

  • A. Farina, E. Valdinoci, On partially and globally overdetermined problems of elliptic type, American Journal of Mathematics, 135 (2013) pp. 1699--1726.

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

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

  • P.-É. Druet, O. Klein, J. Sprekels, F. Tröltzsch, I. Yousept, Optimal control of three-dimensional state-constrained induction heating problems with nonlocal radiation effects, SIAM Journal on Control and Optimization, 49 (2011) pp. 1707--1736.
    Abstract
    The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove the existence and uniqueness of the solution to the state equation. The regularity analysis associated with the time harmonic Maxwell equations is also studied. In the second part of the paper, the existence and uniqueness of the solution to the corresponding linearized equation is shown. With this result at hand, the differentiability of the control-to-state mapping operator associated with the state equation is derived. Finally, based on the theoretical results, first oder necessary optimality conditions for an associated optimal control problem are established.

  • K. Kostourou, D. Peschka, A. Münch, B. Wagner, S. Herminghaus, R. Seemann, Interface morphologies in liquid/liquid dewetting, Chemical Engineering and Processing, 50 (2011) pp. 531--536.
    Abstract
    The dynamics and morphology of a liquid polystyrene (PS) film on the scale of a hundred nanometer dewetting from a liquid polymethylmethacrylate (PMMA) film is investigated experimentally and theoretically. The polymers considered here are both below their entanglement lengths and have negligible elastic properties. A theoretical model based on viscous Newtonian flow for both polymers is set up from which a system of coupled lubrication equations is derived and solved numerically. A direct comparison of the numerical solution with the experimental findings for the characteristic signatures of the cross-sections of liquid/air and liquid/liquid phase boundaries of the dewetting rims as well as the dewetting rates is performed and discussed for various viscosity ratios of the PS and PMMA layers.

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

  • D. Peschka, A. Münch, B. Niethammer, Self-similar rupture of viscous thin films in the strong-slip regime, Nonlinearity, 23 (2010) pp. 409--427.
    Abstract
    We consider rupture of thin viscous films in the strong-slip regime with small Reynolds numbers. Numerical simulations indicate that near the rupture point viscosity and van-der-Waals forces are dominant and that there are self-similar solutions of the second kind. For a corresponding simplified model we rigorously analyse self-similar behaviour. There exists a one-parameter family of self-similar solutions and we establish necessary and sufficient conditions for convergence to any self-similar solution in a certain parameter regime. We also present a conjecture on the domains of attraction of all self-similar solutions which is supported by numerical simulations.

  • P. Krejčí, E. Rocca, J. Sprekels, A bottle in a freezer, SIAM Journal on Mathematical Analysis, 41 (2009) pp. 1851-1873.

  • O. Klein, F. Luterotti, R. Rossi, Existence and asymptotic analysis of a phase field model for supercooling, Quarterly of Applied Mathematics, 64 (2006) pp. 291-319.

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

  • O. Klein, P. Philip, J. Sprekels, Modeling and simulation of sublimation growth of SiC bulk single crystals, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 6 (2004) pp. 295--314.

  • O. Klein, P. Philip, J. Sprekels, K. Wilmanski, Radiation- and convection-driven transient heat transfer during sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 222 (2001) pp. 832--851.

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

  Contributions to Collected Editions

  • D. Peschka, Numerics of contact line motion for thin films, in: 8th Vienna International Conference on Mathematical Modelling -- MATHMOD 2015, 48 of IFAC-PapersOnLine, Elsevier, 2015, pp. 390--393.

  • M.H. Farshbaf Shaker, T. Fukao, N. Yamazaki, Singular limit of Allen--Cahn equation with constraints and its Lagrange multiplier, in: Dynamical Systems, Differential Equations and Applications, AIMS Proceedings 2015, Proceedings of the 10th AIMS International Conference (Madrid, Spain), M. DE León, W. Feng, Z. Feng, X. Lu, J.M. Martell, J. Parcet, D. Peralta-Salas, W. Ruan, eds., AIMS Proceedings, American Institute of Mathematical Sciences, 2015, pp. 418--427.
    Abstract
    We consider the Allen-Cahn equation with constraint. Our constraint is the subdifferential of the indicator function on the closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier to our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our problem.

  • L. Blank, M.H. Farshbaf Shaker, C. Hecht, J. Michl, Ch. Rupprecht, Optimal control of Allen--Cahn systems, in: Trends in PDE Constrained Optimization, G. Leugering, P. Benner ET AL., eds., 165 of International Series of Numerical Mathematics, Birkhäuser, Basel et al., 2014, pp. 11--26.

  • L. Blank, M.H. Farshbaf Shaker, H. Garcke, Ch. Rupprecht, V. Styles, Multi-material phase field approach to structural topology optimization, in: Trends in PDE Constrained Optimization, G. Leugering, P. Benner ET AL., eds., 165 of International Series of Numerical Mathematics, Birkhäuser, Basel et al., 2014, pp. 231--246.

  • P. Rudolph, Ch. Frank-Rotsch, F.-M. Kiessling, W. Miller, U. Rehse, O. Klein, Ch. Lechner, J. Sprekels, B. Nacke, H. Kasjanov, P. Lange, M. Ziem, B. Lux, M. Czupalla, O. Root, V. Trautmann, G. Bethin, Crystal growth in heater-magnet modules --- From concept to use, in: Proceedings of the International Scientific Colloquium Modelling for Electromagnetic Processing (MEP2008), Hannover, October 26--29, 2008, E. Braake, B. Nacke, eds., Leibniz University of Hannover, 2008, pp. 79-84.

  Preprints, Reports, Technical Reports

  • P. Colli, G. Gilardi, J. Sprekels, On a Cahn--Hilliard system with convection and dynamic boundary conditions, Preprint no. 2391, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2391 .
    Abstract, PDF (291 kByte)
    This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn--Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn--Hilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo--Galerkin scheme, is introduced and rigorously discussed.

  • E. Meca Álvarez, A. Münch, B. Wagner, Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity, Preprint no. 2387, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2387 .
    Abstract, PDF (801 kByte)
    We study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material flux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material flux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the time-dependent and non-uniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a flat moving front. The second instability is related to the Mullins-Sekerka instability in the presence of elasticity and arises at the moving front between the two phases when the flux is reversed. Stability analyses of the full model and the corresponding sharp-interface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries.

  • S. Bergmann, D.A. Barragan-Yani, E. Flegel, K. Albe, B. Wagner, Anisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model, Preprint no. 2386, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2386 .
    Abstract, PDF (1682 kByte)
    We present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid-solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger-Weber interatomic potential. The temperature-dependent interface velocity follows a Vogel-Fulcher type behavior and allows to properly account for the dynamics in the undercooled melt.

  • P. Krejčí, E. Rocca, J. Sprekels, Unsaturated deformable porous media flow with phase transition, Preprint no. 2384, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2384 .
    Abstract, PDF (296 kByte)
    In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cut-off techniques and suitable Moser-type estimates.

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

  • M. Hintermüller, M. Hinze, Ch. Kahle, T. Keil, A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system, Preprint no. 2311, WIAS, Berlin, 2016.
    Abstract, PDF (640 kByte)
    This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn--Hilliard--Navier--Stokes system with variable densities. The free energy density associated to the Cahn--Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier--Stokes equation. A dual-weighed residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residual weighted by approximate dual quantities and vice versa as well as various complementary mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.

  • E. Meca Álvarez, A. Münch, B. Wagner, Sharp-interface formation during lithium intercalation into silicon, Preprint no. 2257, WIAS, Berlin, 2016.
    Abstract, PDF (518 kByte)
    In this study we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as a-Si. The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface in the bulk of the layer intersects the free boundary. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

  • A. Münch, B. Wagner, L.P. Cook, R.R. Braun, Apparent slip for an upper convected Maxwell fluid, Preprint no. 2255, WIAS, Berlin, 2016.
    Abstract, PDF (367 kByte)
    In this study the flow field of a nonlocal, diffusive upper convected Maxwell (UCM) fluid with a polymer in a solvent undergoing shearing motion is investigated for pressure driven planar channel flow and the free boundary problem of a liquid layer on a solid substrate. For large ratios of the zero shear polymer viscosity to the solvent viscosity, it is shown that channel flows exhibit boundary layers at the channel walls. In addition, for increasing stress diffusion the flow field away from the boundary layers undergoes a transition from a parabolic to a plug flow. Using experimental data for the wormlike micelle solutions CTAB/NaSal and CPyCl/NaSal, it is shown that the analytic solution of the governing equations predicts these signatures of the velocity profiles. Corresponding flow structures and transitions are found for the free boundary problem of a thin layer sheared along a solid substrate. Matched asymptotic expansions are used to first derive sharp-interface models describing the bulk flow with expressions for an em apparent slip for the boundary conditions, obtained by matching to the flow in the boundary layers. For a thin film geometry several asymptotic regimes are identified in terms of the order of magnitude of the stress diffusion, and corresponding new thin film models with a slip boundary condition are derived.

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

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

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

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

  • T. Ahnert, A. Münch, B. Niethammer, B. Wagner, Stability of concentrated suspensions under Couette and Poiseuille flow, Preprint no. 2159, WIAS, Berlin, 2015.
    Abstract, PDF (678 kByte)
    The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated and connections to the stability problem for the related classical Bingham-flow problem are discussed.

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

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

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

  • S. Dipierro, O. Savin, E. Valdinoci, Boundary behavior of nonlocal minimal surfaces, Preprint no. 2129, WIAS, Berlin, 2015.
    Abstract, PDF (545 kByte)
    We consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical minimal surfaces. In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough. When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturba- tions of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation. In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest.

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

  Talks, Poster

  • D. Peschka, Motion of thin droplets over surfaces, Making a Splash -- Driplets, Jets and Other Singularities, March 20 - 24, 2017, Brown University, Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, USA, March 22, 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. Hintermüller, Non-smooth structures in PDE-constrained optimization, Mathematisches Kolloquium, Universität Duisburg-Essen, Fakultät für Mathematik, Essen, January 11, 2017.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, Kolloquium, Friedrich-Alexander-Universität Erlangen-Nürnberg, Department Mathematik, Erlangen, May 2, 2017.

  • M. Hintermüller, Recent trends in PDE-constrained optimization with non-smooth structures, Fourth Conference on Numerical Analysis and Optimization (NAOIV-2017), January 2 - 5, 2017, Sultan Qaboos University, Muscat, Oman, January 4, 2017.

  • D. Peschka, A free boundary problem for the flow of viscous liquid bilayers, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 26, 2016.

  • D. Peschka, A free boundary problem for the motion of viscous liquids, 7th European Congress of Mathematics (7ECM), Minisymposium 29 ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18 - 22, 2016, Technische Universität Berlin, July 22, 2016.

  • D. Peschka, Multi-phase flows with contact lines: Solid vs liquid substrates, Industrial and Applied Mathematics Seminar, University of Oxford, Mathematical Institute, UK, October 27, 2016.

  • D. Peschka, Thin film free boundary problems --- Modeling of contact line dynamics with gradient formulations, CeNoS-Kolloquium, Westfälische Wilhelms-Universität Münster, Center for Nonlinear Science, January 12, 2016.

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

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

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

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

  • E. Valdinoci, Nonlocal equations from 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, Justus-Liebig-Universität Gießen, Fakultät für Mathematik, February 10, 2016.

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

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

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

  • M.H. Farshbaf Shaker, A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media, International INdAM Conference ``Optimal Control for Evolutionary PDEs and Related Topics (OCERTO 2016)'', June 20 - 24, 2016, Cortona, Italy, June 21, 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, Nonsmooth structures in PDE constrained optimization, 66th Workshop ``Advances in Convex Analysis and Optimization'', July 5 - 10, 2016, International Centre for Scientific Culture ``E. Majorana'', School of Mathematics ``G. Stampacchia'', Erice, Italy, July 9, 2016.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, May 11, 2016.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, The Fifth International Conference on Continuous Optimization, Session: ``Recent Developments in PDE-constrained Optimization I'', August 6 - 11, 2016, Tokyo, Japan, August 10, 2016.

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

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

  • 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.

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

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

  • E. Cinti, Quantitative 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 Cahn--Hilliard equation by the velocity, Numerical Analysis Seminars, Durham University, UK, March 13, 2015.

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

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

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

  • M. Heida, Modeling of fluid interfaces, Jahrestagung der Deutschen Mathematiker-Vereinigung, Minisymposium ``Mathematics of Fluid Interfaces'', September 21 - 25, 2015, Universität Hamburg, Fakultät für Mathematik, Informatik und Naturwissenschaften, Hamburg, September 23, 2015.

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

  • D. Peschka, Modeling and applications of bilayer flows, Seminar of the Research Training Group GRK 1276 ``Structure Formation and Transport in Complex Systems'', Universität des Saarlandes, Institut für Theoretische Physik, Saarbrücken, January 27, 2015.

  • D. Peschka, Numerics of contact line motion for thin films, MATHMOD 2015, Minisymposium ``Free Boundary Problems in Applications: Recent Advances in Modelling, Simulation and Optimization'', February 17 - 20, 2015, Technische Universität Wien, Institut für Analysis und Scientific Computing, Wien, Austria, February 19, 2015.

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

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

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

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

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

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

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

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

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

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

  • D. Hömberg, Nucleation, growth, and grain size evolution in multiphase materials, INdAM Workshop ``Special Materials in Complex Systems -- SMaCS 2015'', May 18 - 22, 2015, Rome, Italy, May 21, 2015.

  • 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  • M.H. Farshbaf Shaker, A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles, Conference on Partial Differential Equations, May 28 - 31, 2014, Novacella, Italy, May 29, 2014.

  • M.H. Farshbaf Shaker, Relating phase field and sharp interface approaches to structural topology optimization, SADCO-WIAS Young Researcher Workshop, January 29 - 31, 2014, WIAS, January 31, 2014.

  • TH. Petzold, On the sharp interface limit of a phase field model with mechanical effects applied to phase transitions in steel, Universität Bremen, Zentrum für Technomathematik, May 22, 2013.

  • O. Klein, Modeling and numerical simulation of the application of traveling magneticfields to stabilize crystal growth from the melt, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11 - 15, 2012, Frauenchiemsee, June 11, 2012.

  • J. Sprekels, A time discretization for a nonstandard viscous Cahn--Hilliard system, INDAM Workshop PDEs for Multiphase Advanced Materials (ADMAT2012), September 17 - 21, 2012, Cortona, Italy, September 19, 2012.

  • J. Sprekels, Mathematical challenges in the industrial growth of semiconductor bulk single crystals, Kick-off Symposium Mathematics: Official Opening of the Felix Klein Building, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, January 13, 2012.

  • 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.

  • J. Sprekels, Mathematical challenges in the industrial growth of semiconductor bulk single crystals, Multiphase and Multiphysics Problems, September 25 - 30, 2011, Riemann International School of Mathematics, Verbania, Italy, September 26, 2011.

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

  • W. Dreyer, Phase transitions and kinetic relations, Séminaire Fluides Compressibles, Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, France, September 30, 2009.

  • W. Dreyer, Phase transitions during hydrogen storage and in lithium-ion batteries, EUROTHERM Seminar no. 84: Thermodynamics of Phase Changes, May 25 - 27, 2009, Université Catholique de Louvain, Namur, Belgium, May 27, 2009.

  • J. Sprekels, Problems in the industrial growth of semiconductor crystals: Radiative transfer, convection, magnetic fields, and free boundaries, Seminar of the Institute of Applied Mathematics and Information Technology, Università di Pavia, Dipartimento di Matematica ``F. Casorati'', Italy, September 23, 2009.

  External Preprints

  • L. Caffarelli, X. Ros-Oton, J. Serra, Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries, Preprint no. arXiv:1601.05843, Cornell University Library, arXiv.org, 2016.
    Abstract
    We study the obstacle problem for integro-differential operators of order $2s$, with $s in (0; 1)$. Our main results establish the optimal $C^(1;s)$ regularity of solutions $u$, and the $C^(1;alpha)$ regularity of the free boundary near regular points. Namely, we prove the following dichotomy at all free boundary points $x0 in partial u = varphi$: (i) either $u(x) -varphi (x) = c d^(1+s)(x) + o( x - x0 ^(1+s+alpha))$ for some $c > 0$, (ii) or u(x) -varphi(x) = o( x - x0 ^(1+s+alpha))$, where $d$ is the distance to the contact set $u - varphi$. Moreover, we show that the set of free boundary points $x0$ satisfying (i) is open, and that the free boundary is $C^(1;alpha)$ near those points. These results were only known for the fractional Laplacian [CSS08], and are completely new for more general integro-di erential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type for- mula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-di erential operators: we establish similar regularity results for obstacle problems with convex operators.

  • X. Ros-Oton, J. Serra, The structure of the free boundary in the fully nonlinear thin obstacle problem, Preprint no. arXiv:1603.09292, Cornell University Library, arXiv.org, 2016.
    Abstract
    We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Our main result establishes that the free boundary is C$^1$ near regular points.

  • D.A. Gomes, S. Patrizi, Obstacle mean-field game problem, Preprint no. arXiv:1410.6942, Cornell University Library, arXiv.org, 2014.
    Abstract
    In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions.