Im Zusammenhang mit freien Randwertproblemen werden am WIAS Themen wie Eigenschaften von Lösungen, PhasenfeldApproximationen, Kompatibilität mit der Thermodynamik, Beschreibung dünner Filme, Variationelle Ungleichungen, (implizite) Hindernisprobleme und Anwendungen beim Warmformen behandelt.
Ausführlichere Darstellungen der WIASForschungsthemen finden sich auf der jeweils zugehörigen englischen Seite.
Publikationen
Monografien

M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., NonSmooth and ComplementarityBased Distributed Parameter Systems: Simulation and Hierarchical Optimization, 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, viii + 519 pages, (Collection Published), DOI 10.1007/9783030793937 .

M. Hintermüller, J.F. Rodrigues, eds., Topics in Applied Analysis and Optimisation  Partial Differential Equations, Stochastic and Numerical Analysis, CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, 396 pages, (Collection Published).

M. Hintermüller, D. Wegner, Distributed and Boundary Control Problems for the Semidiscrete CahnHilliard/NavierStokes System with Nonsmooth GinzburgLandau Energies, in: Topological Optimization and Optimal Transport in the Applied Sciences, M. Bergounioux, E. Oudet, M. Rumpf, G. Carlier, Th. Champion, F. Santambrogio, eds., 17 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2017, pp. 4063, (Chapter Published).

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).

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).
Artikel in Referierten Journalen

L. Schmeller, D. Peschka, Sharpinterface limits of CahnHilliard models and mechanics with moving contact lines, , 22 (2024), pp. 869890, DOI 10.1137/23M1546592 .
Abstract
We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phasefield models converge to certain sharpinterface models when the interface thickness tends to zero. In particular, we study the scaling of the CahnHilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharpinterface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces. 
D. Peschka, Partial and complete wetting of thin films with dynamic contact angle, Physics of Fluids, 35 (2023), pp. 041705/1041705/6, DOI 10.1063/5.0146538 .
Abstract
The wetting of thin films depends critically on the sign of the spreading coefficient S. We discuss the cases S<0, S=0, and S>0 for transient models with contact line dissipation and find that the use of a dynamic contact angle solves problems for S>0 that models might otherwise have. For initial data with a nonzero slope and S>0, we show that there exists a finite time at which the contact angle of the thin film goes to zero. Then, a molecular precursor emerges from the thin film and moves outward at a constant velocity. 
A.K. Barua, R. Chew, L. Shuwang, J. Lowengrub, A. Münch, B. Wagner, Sharpinterface problem of the OhtaKawasaki model for symmetric diblock copolymers, Journal of Computational Physics, 481 (2023), pp. 112032/1112032/23, DOI 10.1016/j.jcp.2023.112032 .
Abstract
The OhtaKawasaki model for diblockcopolymers is well known to the scientific community of diffuseinterface methods. To accurately capture the longtime evolution of the moving interfaces, we present a derivation of the corresponding sharpinterface limit using matched asymptotic expansions, and show that the limiting process leads to a HeleShaw type moving interface problem. The numerical treatment of the sharpinterface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the OhtaKawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multiconnected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular farfield boundary. The integral formulation of the problem involves both double and singlelayer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a nonequilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space. 
L. Giacomelli, M. Gnann, D. Peschka, Droplet motion with contactline friction: Longtime asymptotics in complete wetting, Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 479 (2023), pp. 20230090/120230090/23, DOI 10.1098/rspa.2023.0090 .
Abstract
We consider the thinfilm equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal longtime asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasiselfsimilar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem. 
M. Hintermüller, T. Keil, Strong stationarity conditions for the optimal control of a CahnHilliardNavierStokes system, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 89 (2024), pp. 12/112/28 (published online on 05.12.2023), DOI 10.1007/s00245023100639 .
Abstract
This paper is concerned with the distributed optimal control of a timediscrete CahnHilliardNavierStokes system with variable densities. It focuses on the doubleobstacle potential which yields an optimal control problem for a variational inequality of fourth order and the NavierStokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel. 
A.H. Erhardt, E. Wahlén, J. Weber, Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl, Studies in Applied Mathematics, 149 (2022), pp. 904942, DOI 10.1111/sapm.12525 .
Abstract
We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinateinduced singularities and to cast the problem in the form ?identity plus compact,? which is amenable to Rabinowitz's global bifurcation theorem, whereas no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed. 
A.K. Giri, P. Malgaretti, D. Peschka, M. Sega, Resolving the microscopic hydrodynamics at the moving contact line, Physical Review Fluids, 7 (2022), pp. L102001/1L102001/9, DOI 10.1103/PhysRevFluids.7.L102001 .
Abstract
By removing the smearing effect of capillary waves in molecular dynamics simulations we are able to provide a microscopic picture of the region around the moving contact line (MCL) at an unprecedented resolution. On this basis, we show that the continuum character of the velocity field is unaffected by molecular layering down to below the molecular scale. The solution of the continuum Stokes problem with MCL and Navierslip matches very well the molecular dynamics data and is consistent with a sliplength of 42 Å and small contact line dissipation. This is consistent with observations of the local force balance near the liquidsolid interface. 
A. Alphonse, C.N. Rautenberg, J.F. Rodrigues, Analysis of a quasivariational contact problem arising in thermoelasticity, Nonlinear Analysis. An International Mathematical Journal, 217 (2022), pp. 112728/1112728/40, DOI 10.1016/j.na.2021.112728 .
Abstract
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membranemould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasivariational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under nondegenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semidiscretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of timedependent solutions. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, On the differentiability of the minimal and maximal solution maps of elliptic quasivariational inequalities, Journal of Mathematical Analysis and Applications, 507 (2022), pp. 125732/1125732/19, DOI 10.1016/j.jmaa.2021.125732 .
Abstract
In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasivariational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, Optimal control and directional differentiability for elliptic quasivariational inequalities, SetValued and Variational Analysis. Theory and Applications. Springer, Dordrecht. English., 30 (2022), pp. 873922, DOI 10.1007/s1122802100624x .
Abstract
We focus on elliptic quasivariational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general unsigned data, thereby extending the results of our previous work which provided a first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area. 
G. Dong, M. Hintermüller, Y. Zhang, A class of secondorder geometric quasilinear hyperbolic PDEs and their application in imaging, SIAM Journal on Imaging Sciences, 14 (2021), pp. 645688, DOI 10.1137/20M1366277 .
Abstract
In this paper, we study damped secondorder dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of secondorder damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped secondorder total variation flow, which is primarily motivated by the application of image denoising; the other is a damped secondorder mean curvature flow for level sets of scalar functions, which is related to a nonconvex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some secondorder geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with firstorder flows are also documented. 
M. Thomas, S. Tornquist, Discrete approximation of dynamic phasefield fracture in viscoelastic materials, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 38653924, DOI 10.3934/dcdss.2021067 .
Abstract
This contribution deals with the analysis of models for phasefield fracture in viscoelastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rateindependent law with a positively 1homogeneous dissipation potential. Both evolution laws encode a nonsmooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional. 
A. Alphonse, C.N. Rautenberg, J.F. Rodrigues, Analysis of a quasivariational contact problem arising in thermoelasticity, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 217 (2022), pp. 112728/1112728/40 (published online on 13.12.2021), DOI 10.1016/j.na.2021.112728 .
Abstract
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membranemould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasivariational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under nondegenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semidiscretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of timedependent solutions. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, On the differentiability of the minimal and maximal solution maps of elliptic quasivariational inequalities, Journal of Mathematical Analysis and Applications, published online on 27.10.2021, DOI 10.1016/j.jmaa.2021.125732 .
Abstract
In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasivariational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities. 
N.A. Dao, J.I. Díaz, Q.B.H. Nguyen, Pointwise gradient estimates in multidimensional slow diffusion equations with a singular quenching term, Advanced Nonlinear Studies, 20 (2020), pp. 373384, DOI 10.1515/ans20202076 .

R. Rossi, U. Stefanelli, M. Thomas, Rateindependent evolution of sets, Discrete and Continuous Dynamical Systems  Series S, 14 (2021), pp. 89119 (published online in March 2020), DOI 10.3934/dcdss.2020304 .
Abstract
The goal of this work is to analyze a model for the rateindependent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given timedependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated timeincremental minimization scheme. In the brittle case, this timediscretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energydissipation balance in the timecontinuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, Existence, iteration procedures and directional differentiability for parabolic QVIs, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 95/195/53, DOI 10.1007/s00526020017326 .
Abstract
We study parabolic quasivariational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator. 
G. Dong, H. Guo, Parametric polynomial preserving recovery on manifolds, SIAM Journal on Scientific Computing, 42 (2020), pp. A1885A1912, DOI 10.1137/18M1191336 .

D. Peschka, S. Haefner, L. Marquant, K. Jacobs, A. Münch, B. Wagner, Signatures of slip in dewetting polymer films, Proceedings of the National Academy of Sciences of the United States of America, 116 (2019), pp. 92759284, DOI 10.1073/pnas.1820487116 .

E. Meca Álvarez, A. Münch, B. Wagner, Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 97 (2018), pp. 012801/1012801/12, DOI 10.1103/PhysRevE.97.012801 .
Abstract
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 timedependent and nonuniform 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 MullinsSekerka 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 sharpinterface 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. 
A. Ceretani, C.N. Rautenberg, The Boussinesq system with mixed nonsmooth boundary conditions and ``donothing'' boundary flow, Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 70 (2019), pp. 14/114/24 (published online on 07.12.2018), DOI 10.1007/s000330181058y .
Abstract
A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel nonsmooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the donothing boundary condition for the fluid: 1) the directional donothing condition and 2) the donothing condition together with an integral bound for the backflow. Wellposedness of variational formulations is proved for each problem. 
T. Ahnert, A. Münch, B. Niethammer, B. Wagner, Stability of concentrated suspensions under Couette and Poiseuille flow, Journal of Engineering Mathematics, 111 (2018), pp. 5177, DOI 10.1007/s106650189954x .
Abstract
The stability of twodimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the twophase 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 twophase flow model may become illposed as the particle phase approaches its maximum packing fraction. The case of twodimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shearinduced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Binghamtype flow is investigated and connections to the stability problem for the related classical Binghamflow problem are discussed. 
T. Ahnert, A. Münch, B. Wagner, Models for the twophase flow of concentrated suspensions, European Journal of Applied Mathematics, 30 (2019), pp. 585617 (published online on 04.06.2018), DOI 10.1017/S095679251800030X .
Abstract
A new twophase model is derived that make use of a constitutive law combining nonBrownian suspension with granular rheology, that was recently proposed by Boyer et al. [PRL, 107(18),188301 (2011)]. It is shown that for the simple channel flow geometry, the stress model naturally exhibits a Bingham type flow property with an unyielded finitesize zone in the center of the channel. As the volume fraction of the solid phase is increased, the various transitions in the flow fields are discussed using phase space methods for a boundary value problem, that is derived from the full model. The predictions of this analysis is then compared to the direct finiteelement numerical solutions of the full model. 
H. Antil, C.N. Rautenberg, Fractional elliptic quasivariational inequalities: Theory and numerics, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 20 (2018), pp. 124, DOI 10.4171/IFB/395 .

C. Bertoglio, A. Caiazzo, Y. Bazilevs, M. Braack, M. EsmailyMoghadam, V. Gravemeier, A.L. Marsden, O. Pironneau, I.E. VignonClementel, W.A. Wall, Benchmark problems for numerical treatment of backflow at open boundaries, International Journal of Numerical Methods in Biomedical Engineering, 34 (2018), pp. e2918/1e2918/34, DOI 10.1002/cnm.2918 .
Abstract
In computational fluid dynamics, incoming velocity at open boundaries, or backflow, often yields to unphysical instabilities already for moderate Reynolds numbers. Several treatments to overcome these backflow instabilities have been proposed in the literature. However, these approaches have not yet been compared in detail in terms of accuracy in different physiological regimes, in particular due to the difficulty to generate stable reference solutions apart from analytical forms. In this work, we present a set of benchmark problems in order to compare different methods in different backflow regimes (with a full reversal flow and with propagating vortices after a stenosis). The examples are implemented in FreeFem++ and the source code is openly available, making them a solid basis for future method developments. 
P. Colli, G. Gilardi, J. Sprekels, On a CahnHilliard system with convection and dynamic boundary conditions, Annali di Matematica Pura ed Applicata. Serie Quarta. Fondazione Annali di Matematica Pura ed Applicata, c/o Dipartimento di Matematica ``U. Dini'', Firenze; SpringerVerlag, Heidelberg. English, French, German, Italian, English abstracts., 197 (2018), pp. 14451475, DOI 10.1007/s1023101807321 .
Abstract
This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of CahnHilliard 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 CahnHilliard 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 FaedoGalerkin scheme, is introduced and rigorously discussed. 
G. Kitavtsev, A. Münch, B. Wagner, Thin film models for an active gel, Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 474 (2018), pp. 20170828/120170828/20, DOI 10.1098/rspa.2017.0828 .

M. Hintermüller, M. Hinze, Ch. Kahle, T. Keil, A goaloriented dualweighted adaptive finite element approach for the optimal control of a nonsmooth CahnHilliardNavierStokes system, Optimization and Engineering. International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences, 19 (2018), pp. 629662, DOI 10.1007/s1108101893936 .
Abstract
This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a timediscrete CahnHilliardNavierStokes system with variable densities. The free energy density associated to the CahnHilliard system incorporates the doubleobstacle 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 NavierStokes equation. A dualweighed residual approach for goaloriented adaptive finite elements is presented which is based on the concept of Cstationarity. 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. 
S. Bergmann, D.A. BarraganYani, E. Flegel, K. Albe, B. Wagner, Anisotropic solidliquid interface kinetics in silicon: An atomistically informed phasefield model, Modelling and Simulation in Materials Science and Engineering, 25 (2017), pp. 065015/1065015/20, DOI 10.1088/1361651X/aa7862 .
Abstract
We present an atomistically informed parametrization of a phasefield model for describing the anisotropic mobility of liquidsolid 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 StillingerWeber interatomic potential. The temperaturedependent interface velocity follows a VogelFulcher type behavior and allows to properly account for the dynamics in the undercooled melt. 
W. Dreyer, C. Guhlke, Sharp limit of the viscous CahnHilliard equation and thermodynamic consistency, Continuum Mechanics and Thermodynamics, 29 (2017), pp. 913934.
Abstract
Diffuse and sharp interface models represent two alternatives to describe phase transitions with an interface between two coexisting phases. The two model classes can be independently formulated. Thus there arises the problem whether the sharp limit of the diffuse model fits into the setting of a corresponding sharp interface model. We call a diffuse model admissible if its sharp limit produces interfacial jump conditions that are consistent with the balance equations and the 2nd law of thermodynamics for sharp interfaces. We use special cases of the viscous CahnHilliard equation to show that there are admissible as well as nonadmissible diffuse interface models. 
M. Dziwnik, A. Münch, B. Wagner, An anisotropic phasefield model for solidstate dewetting and its sharpinterface limit, Nonlinearity, 30 (2017), pp. 14651496.
Abstract
We propose a phase field model for solid state dewetting in form of a CahnHilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the filmsubstrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. The resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the natural boundary conditions, as indicated from the first variation of the total free energy, imply a contact angle condition for the dewetting front, which, in the isotropic case, is consistent with the wellknown Young's equation. 
H. Antil, M. Hintermüller, R.H. Nochetto, Th.M. Surowiec, D. Wegner, Finite horizon model predictive control of electrowetting on dielectric with pinning, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 19 (2017), pp. 130, DOI 10.4171/IFB/375 .

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 
P. Krejčí, E. Rocca, J. Sprekels, Unsaturated deformable porous media flow with thermal phase transition, Mathematical Models & Methods in Applied Sciences, 27 (2017), pp. 26752710, DOI 10.1142/S0218202517500555 .
Abstract
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 liquidsolid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the ClausiusDuhem 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 cutoff techniques and suitable Mosertype estimates. 
A. Münch, B. Wagner, L.P. Cook, R.R. Braun, Apparent slip for an upper convected Maxwell fluid, SIAM Journal on Applied Mathematics, 77 (2017), pp. 537564, DOI 10.1137/16M1056869 .
Abstract
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 sharpinterface 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. 
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. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasivariational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017), pp. 135.
Abstract
A class of abstract nonlinear evolution quasivariational 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 semidiscrete 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 gradienttype. 
E. Meca Álvarez, A. Münch, B. Wagner, Sharpinterface formation during lithium intercalation into silicon, European Journal of Applied Mathematics, 29 (2018), pp. 118145, DOI 10.1017/S0956792517000067 .
Abstract
In this study we present a phasefield model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as aSi. The governing equations couple a viscous CahnHilliardReaction model with elasticity in the framework of the CahnLarché 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 sharpinterface 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 sharpinterface model with the longtime dynamics of the phasefield model. 
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. 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. 
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. 11671187.
Abstract
Anisotropy is an essential feature of phasefield 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 phasefield 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  GinzburgLandau type phasefield model for dendritic growth. For the resulting derivativefree 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 ButlerVolmer equation in nonequilibrium thermodynamics, Physical Chemistry Chemical Physics, 18 (2016), pp. 2496624983, DOI 10.1039/C6CP04142F .
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 ButlerVolmer equation which has its origin in kinetic theories. The ButlerVolmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the ButlerVolmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the ButlerVolmer equation. Based on nonequilibrium thermodynamics we have recently derived a reduced model for the electrodeelectrolyte 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 ButlerVolmer equation. We discuss the application of the new ButlerVolmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithiumironphosphate 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 CahnHilliard/NavierStokes 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 NavierStokes system with a convective nonlocal CahnHilliard equation in two dimensions of space. We apply recently proved wellposedness and regularity results in order to establish existence of optimal controls as well as firstorder 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 dualitybased pathfollowing semismooth Newton method for elastoplastic contact problems, Journal of Computational and Applied Mathematics, 292 (2016), pp. 150173.

M. Hintermüller, Th. Surowiec, A bundlefree implicit programming approach for a class of elliptic MPECs in function space, Mathematical Programming Series A, 160 (2016), pp. 271305.

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. 
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. 
P.É. Druet, Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some lowfrequency Maxwell equations, Discrete and Continuous Dynamical Systems, 8 (2015), pp. 479496.
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 DivCurl Lemmatype argument provides compact embedding results for such vector fields. We investigate the regularity of the solution fields for the lowfrequency approximation of the Maxwell equations in timeharmonic 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. 24432463.
Abstract
We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solidliquid interface in a twophase 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 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. 
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. Servadei, E. Valdinoci, The BrezisNirenberg result for the fractional Laplacian, Transactions of the American Mathematical Society, 367 (2015), pp. 67102.

L. Caffarelli, O. Savin , E. Valdinoci, Minimization of a fractional perimeterDirichlet integral functional, Annales de l'Institut Henri Poincare. Analyse Non Lineaire, 32 (2015), pp. 901924.

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.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 ``U. 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. 
M.G. Hennessy, V.M. Burlakov, A. Münch, B. Wagner, A. Goriely, Controlled topological transitions in thinfilm phase separation, SIAM Journal on Applied Mathematics, 75 (2015), pp. 3860.
Abstract
In this paper the evolution of a binary mixture in a thinfilm 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 phasefield model and using matched asymptotic expansions a corresponding sharpinterface problem for the location of the interface is established. It is then argued that a thus created twolayer system is not at its energetic minimum but destabilizes into a controlled selfreplicating 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 thinfilm model for the free interfaces, which is derived asymptotically from the sharpinterface model. 
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.

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

G. Aki, W. Dreyer, J. Giesselmann, Ch. Kraus, A quasiincompressible diffuse interface model with phase transition, Mathematical Models & Methods in Applied Sciences, 24 (2014), pp. 827861.
Abstract
This work introduces a new thermodynamically consistent diffuse model for twocomponent 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 NavierStokes equations in the bulk phases equipped with admissible interfacial conditions. For the NavierStokes 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. 629650.

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. 10251058.
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 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.

C. Bertoglio, A. Caiazzo, A tangential regularization method for backflow stabilization in hemodynamics, Journal of Computational Physics, 261 (2014), pp. 162171.
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 threedimensional Womersley flows, as well as considering a hemodynamic physiological regime in a patientspecific 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. 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.

S. Melchionna, E. Rocca, On a nonlocal CahnHilliard equation with a reaction term, Advances in Mathematical Sciences and Applications, 24 (2014), pp. 461497.
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 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.

P.É. Druet, Global Lipschitz continuity for elliptic transmission problems with a boundary intersecting interface, Mathematica Bohemica, 138 (2013), pp. 185224.
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 squareintegrable 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. 16991726.

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. 233258.
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 higherorder 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. 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. 
P.É. Druet, O. Klein, J. Sprekels, F. Tröltzsch, I. Yousept, Optimal control of threedimensional stateconstrained induction heating problems with nonlocal radiation effects, SIAM Journal on Control and Optimization, 49 (2011), pp. 17071736.
Abstract
The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, timeharmonic 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 temperaturedependent 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 controltostate 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. 531536.
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 crosssections 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 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. 
D. Peschka, A. Münch, B. Niethammer, Selfsimilar rupture of viscous thin films in the strongslip regime, Nonlinearity, 23 (2010), pp. 409427.
Abstract
We consider rupture of thin viscous films in the strongslip regime with small Reynolds numbers. Numerical simulations indicate that near the rupture point viscosity and vanderWaals forces are dominant and that there are selfsimilar solutions of the second kind. For a corresponding simplified model we rigorously analyse selfsimilar behaviour. There exists a oneparameter family of selfsimilar solutions and we establish necessary and sufficient conditions for convergence to any selfsimilar solution in a certain parameter regime. We also present a conjecture on the domains of attraction of all selfsimilar 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. 18511873.

O. Klein, F. Luterotti, R. Rossi, Existence and asymptotic analysis of a phase field model for supercooling, Quarterly of Applied Mathematics, 64 (2006), pp. 291319.

W. Dreyer, B. Wagner, Sharpinterface model for eutectic alloys. Part I: Concentration dependent surface tension, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 7 (2005), pp. 199227.

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. 295314.

O. Klein, P. Philip, J. Sprekels, K. Wilmanski, Radiation and convectiondriven transient heat transfer during sublimation growth of silicon carbide single crystals, Journal of Crystal Growth, 222 (2001), pp. 832851.

W. Dreyer, W.H. Müller, A study of the coarsening in tin/lead solders, International Journal of Solids and Structures, 37 (2000), pp. 38413871.
Beiträge zu Sammelwerken

A. Alphonse, M. Hintermüller, C.N. Rautenberg, Stability and sensitivity analysis for quasivariational inequalities, in: NonSmooth and ComplementarityBased Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 183210.

D. Gahururu, M. Hintermüller, S.M. Stengl, Th.M. Surowiec, Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms and risk aversion, in: NonSmooth and ComplementarityBased Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 145181.

C. Grässle, M. Hintermüller, M. Hinze, T. Keil, Simulation and control of a nonsmooth CahnHilliard NavierStokes system with variable fluid densities, in: NonSmooth and ComplementarityBased Distributed Parameter Systems: Simulation and Hierarchical Optimization, M. Hintermüller, R. Herzog, Ch. Kanzow, M. Ulbrich, S. Ulbrich, eds., 172 of International Series of Numerical Mathematics, Birkhäuser, Springer Nature Switzerland AG, Cham, 2022, pp. 211240.

R. Rossi, U. Stefanelli, M. Thomas, Rateindependent evolution of sets, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems  Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 89119, DOI 10.3934/dcdss.2020304 .
Abstract
The goal of this work is to analyze a model for the rateindependent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given timedependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes.In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated timeincremental minimization scheme. In the brittle case, this timediscretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energydissipation balance in the timecontinuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions. 
D. Peschka, M. Thomas, T. Ahnert, A. Münch, B. Wagner, Gradient structures for flows of concentrated suspensions, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 295318, DOI 10.1007/9783030331160 .
Abstract
In this work we investigate a twophase model for concentrated suspensions. We construct a PDE formulation using a gradient flow structure featuring dissipative coupling between fluid and solid phase as well as different driving forces. Our construction is based on the concept of flow maps that also allows it to account for flows in moving domains with free boundaries. The major difference compared to similar existing approaches is the incorporation of a nonsmooth twohomogeneous term to the dissipation potential, which creates a normal pressure even for pure shear flows. 
D. Peschka, Numerics of contact line motion for thin films, in: 8th Vienna International Conference on Mathematical Modelling  MATHMOD 2015, 48 of IFACPapersOnLine, Elsevier, 2015, pp. 390393.

M.H. Farshbaf Shaker, T. Fukao, N. Yamazaki, Singular limit of AllenCahn 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. PeraltaSalas, W. Ruan, eds., AIMS Proceedings, American Institute of Mathematical Sciences, 2015, pp. 418427.
Abstract
We consider the AllenCahn 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 AllenCahn 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. 1126.

L. Blank, M.H. Farshbaf Shaker, H. Garcke, Ch. Rupprecht, V. Styles, Multimaterial 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. 231246.

P. Rudolph, Ch. FrankRotsch, 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 heatermagnet modules  From concept to use, in: Proceedings of the International Scientific Colloquium Modelling for Electromagnetic Processing (MEP2008), Hannover, October 2629, 2008, E. Braake, B. Nacke, eds., Leibniz University of Hannover, 2008, pp. 7984.
Preprints, Reports, Technical Reports

A. Alphonse, D. Caetano, Ch.M. Elliott, Ch. Venkataraman, Free boundary limits of coupled bulksurface models for receptorligand interactions on evolving domains, Preprint no. 3122, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3122 .
Abstract, PDF (5947 kByte)
We derive various novel free boundary problems as limits of a coupled bulksurface reactiondiffusion system modelling ligandreceptor dynamics on evolving domains. These limiting free boundary problems may be formulated as Stefantype problems on an evolving hypersurface. Our results are new even in the setting where there is no domain evolution. The models are of particular relevance to a number of applications in cell biology. The analysis utilises L^{∞}estimates in the manner of De Giorgi iterations and other technical tools, all in an evolving setting. We also report on numerical simulations. 
M. Dambrine, C. Geiersbach, H. Harbrecht, Twonorm discrepancy and convergence of the stochastic gradient method with application to shape optimization, Preprint no. 3121, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3121 .
Abstract, PDF (447 kByte)
The present article is dedicated to proving convergence of the stochastic gradient method in case of random shape optimization problems. To that end, we consider Bernoulli's exterior free boundary problem with a random interior boundary. We recast this problem into a shape optimization problem by means of the minimization of the expected Dirichlet energy. By restricting ourselves to the class of convex, sufficiently smooth domains of bounded curvature, the shape optimization problem becomes strongly convex with respect to an appropriate norm. Since this norm is weaker than the differentiability norm, we are confronted with the socalled twonorm discrepancy, a wellknown phenomenon from optimal control. We therefore need to adapt the convergence theory of the stochastic gradient method to this specific setting correspondingly. The theoretical findings are supported and validated by numerical experiments. 
A. Alphonse, M. Hintermüller, C.N. Rautenberg, G. Wachsmuth, Minimal and maximal solution maps of elliptic QVIs: Penalisation, Lipschitz stability, differentiability and optimal control, Preprint no. 3093, WIAS, Berlin, 2024.
Abstract, PDF (501 kByte)
Quasivariational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the controltostate operator. We also consider a Moreau?Yosidatype penalisation for the QVI wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) Cstationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result. 
J. Li, X. Liu, D. Peschka, Local wellposedness and global stability of onedimensional shallow water equations with surface tension and constant contact angle, Preprint no. 3084, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3084 .
Abstract, PDF (405 kByte)
We consider the onedimensional shallow water problem with capillary surfaces and moving contact lines. An energybased model is derived from the twodimensional water wave equations, where we explicitly discuss the case of a stationary force balance at a moving contact line and highlight necessary changes to consider dynamic contact angles. The moving contact line becomes our free boundary at the level of shallow water equations, and the depth of the shallow water degenerates near the free boundary, which causes singularities for the derivatives and degeneracy for the viscosity. This is similar to the physical vacuum of compressible flows in the literature. The equilibrium, the global stability of the equilibrium, and the local wellposedness theory are established in this paper. 
M.H. Farshbaf Shaker, M. Thomas, Analysis of a compressible Stokesflow with degenerating and singular viscosity, Preprint no. 2786, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2786 .
Abstract, PDF (744 kByte)
In this paper we show the existence of a weak solution for a compressible singlephase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a densitydependent viscosity. The analysis is based on an implicit timediscrete scheme and a Galerkinapproximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of pLaplacian type in the transport equation that allows for refined compactness arguments on subdomains. 
A. Münch, B. Wagner, Selfconsistent field theory for a polymer brush. Part II: The effective chemical potential, Preprint no. 2649, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2649 .
Abstract, PDF (318 kByte)
The most successful meanfield model to describe the collective behaviour of the large class of macromolecular polymers is the selfconsistent field theory (SCFT). Still, even for the simple system of a grafted dry polymer brush, the meanfield equations have to be solved numerically. As one of very few alternatives that offer some analytical tractability the strongstretching theory (SST) has led to explicit expressions for the effective chemical potential and consequently the free energy to promote an understanding of the underlying physics. Yet, a direct derivation of these analytical results from the SCFT model is still outstanding. In this study we present a systematic asymptotic theory based on matched asymtptotic expansions to obtain the effective chemical potential from the SCFT model for a dry polymer brush for large but finite stretching. 
A. Münch, B. Wagner, Selfconsistent field theory for a polymer brush. Part I: Asymptotic analysis in the strongstretching limit, Preprint no. 2648, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2648 .
Abstract, PDF (854 kByte)
In this study we consider the selfconsistent field theory for a dry, in compressible polymer brush, densely grafted on a substrate, describing the average segment density of a polymer in terms of an effective chemical potential for the interaction between the segments of the polymer chain. We present a systematic singular perturbation analysis of the selfconsistent field theory in the strongstretching limit, when the length scale of the ratio of the radius of gyration of the polymer chain to the extension of the brush from the substrate vanishes. Our analysis yields, for the first time, an approximation for the average segment density that is correct to leading order in the outer scaling and resolves the boundary layer singularity at the end of the polymer brush in the strongstretching limit. We also show that in this limit our analytical results agree increasingly well with our numerical solutions to the full model equations comprising the selfconsistent field theory. 
K. Disser, Global existence, uniqueness and stability for nonlinear dissipative systems of bulkinterface interaction, Preprint no. 2313, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2313 .
Abstract, PDF (302 kByte)
We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulkinterface interaction. The setting includes nonsmooth geometries and e.g. slow, fast and "entropic” diffusion processes under mass conservation. The main results are global wellposedness and exponential stability of equilibria. As a part of the proof, we show bulkinterface maximum principles and a bulkinterface 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 AllenCahn dissipative dynamics and to include large classes of inhomogeneous boundary conditions and external forces. 
A. Farina, E. Valdinoci, Anisotropic nonlocal operators regularity and rigidity theorems for a class of anisotropic nonlocal operators, Preprint no. 2213, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2213 .
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. 
V. Barbu, P. Colli, G. Gilardi, G. Marinoschi, E. Rocca, Sliding modes for a phasefield system, Preprint no. 2133, WIAS, Berlin, 2015, DOI 10.20347/WIAS.PREPRINT.2133 .
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. 
W. Dreyer, J. Giesselmann, Ch. Kraus, Modeling of compressible electrolytes with phase transition, Preprint no. 1955, WIAS, Berlin, 2014, DOI 10.20347/WIAS.PREPRINT.1955 .
Abstract, PDF (457 kByte)
A novel thermodynamically consistent diffuse interface model is derived for compressible electrolytes with phase transitions. The fluid mixtures may consist of N constituents with the phases liquid and vapor, where both phases may coexist. In addition, all constituents may consist of polarizable and magnetizable matter. Our introduced thermodynamically consistent diffuse interface model may be regarded as a generalized model of AllenCahn/NavierStokes/Poisson type for multicomponent flows with phase transitions and electrochemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a noncoupled and a coupled regime, where the coupling takes place between the smallness parameter in the Poisson equation and the width of the interface. We recover in the sharp interface limit a generalized AllenCahn/Euler/Poisson system for mixtures with electrochemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satisfy, for instance, a generalized GibbsThomson law and a dynamic YoungLaplace law. 
P.É. Druet, A curvature estimate for open surfaces subject to a general mean curvature operator and natural contact conditions at their boundary, Preprint no. 1897, WIAS, Berlin, 2013, DOI 10.20347/WIAS.PREPRINT.1897 .
Abstract, PDF (251 kByte)
In the seventies, L. Simon showed that the main curvatures of twodimensional hypersurfaces obeying a general equation of mean curvature type are a priori bounded by the Hölder norm of the coefficients of the surface differential operator. This was an essentially interior estimate. In this paper, we provide a complement to the theory, proving a global curvature estimate for open surfaces that satisfy natural contact conditions at the intersection with a given boundary.
Vorträge, Poster

M. Thomas, Analysis of a model for viscoelastoplastic twophase flows in geodynamics, 23rd Symposium on Trends in Applications of Mathematics to Mechanics (STAMM 2024), April 3  5, 2024, JuliusMaximiliansUniversität Würzburg, April 5, 2024.

M. Thomas, Analysis of a model for viscoelastoplastic twophase flows in geodynamics, 9th European Congress of Mathematics (9ECM), Minisymposium 27 ``New Trends in Calculus of Variations'', July 15  19, 2024, Universidad de Sevilla, Spain, July 16, 2024.

M. Thomas, Analysis of a model for viscoelastoplastic twophase flows in geodynamics, Seminar on Nonlinear Partial Differential Equations, Texas A&M University, Department of Mathematics, College Station, USA, March 19, 2024.

A. Alphonse, Riskaverse optimal control of elliptic variational inequalities, 94th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2024), Session 19.01 ``Various topics in Optimization of Differential Equations (1)'', March 18  22, 2024, OttovonGuerickeUniversität Magdeburg, March 19, 2024.

M. Hintermüller, QVIs: Semismooth Newton, optimal control, and uncertainties, RICAM Colloquium, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, June 27, 2024.

M. Hintermüller, Riskaverse optimal control of random elliptic VIs, MS43 2024 SIAM Conference on Uncertainty Quantification (UQ24), Minisymposium MS43: ``Efficient Solution Schemes for Optimization of Complex Systems Under Uncertainty'', February 27  March 1, 2024, Trieste, Italy, February 27, 2024.

D. Peschka, Moving contact lines for sliding droplets, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 11 ``Interfacial Flows'', May 30  June 2, 2023, Technische Universität Dresden, June 1, 2023.

D. Peschka, Multiscale limits of thinfilm models with moving support, In Search of Model Structures for Nonequilibrium Systems, April 24  28, 2023, Westfälische WilhelmsUniversität Münster, April 27, 2023.

D. Peschka, Sharpinterface limit of models with mechanics and contact lines, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Session 00247 ``Interfaces and Free Boundaries in Fluid Mechanics and Materials Science'', August 20  25, 2023, Waseda University, Tokyo, Japan, August 24, 2023.

M. Thomas, Approximating dynamic phasefield fracture with a firstorder formulation for velocity and stress, Annual Workshop of the GAMM Activity Group on Analysis of PDEs, September 18  20, 2023, Katholische Universität EichstättIngolstadt, September 20, 2023.

M. Thomas, Damage in viscoelastic materials at finite strains, Workshop ``Variational Methods for Evolution'', December 3  8, 2023, Mathematisches Forschungsinstitut Oberwolfach, December 7, 2023.

M. Thomas, Some aspects of damage in nonlinearly elastic materials: From damage to delamination in nonlinearly elastic materials, Variational and Geometric Structures for Evolution, October 9  13, 2023, Università Commerciale Luigi Bocconi, Levico Terme, Italy, October 10, 2023.

M. Thomas, Approximating dynamic phasefield fracture with a firstorder formulation for velocity and stress, Nonlinear PDEs: Recent Trends in the Analysis of Continuum Mechanics, July 17  21, 2023, Universität Bonn, Hausdorff School for Advanced Studies in Mathematics, July 17, 2023.

M. Thomas, Approximating dynamic phasefield fracture with a firstorder formulation for velocity and stress, Seminar für Angewandte Mathematik, Technische Universität Dresden, June 5, 2023.

M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, March 27, 2023.

A. Alphonse, Analysis of a quasivariational contact problem arising in thermoelasticity, European Conference on Computational Optimization (EUCCO), Session ``Nonsmooth Optimization'', September 25  27, 2023, Universität Heidelberg, September 25, 2023.

M. Hintermüller, Optimal control of multiphase fluids and droplets (online talk), Workshop ``Control Methods in Hyperbolic Partial Differential Equations'' (Hybrid Event), November 5  10, 2023, Mathematisches Forschungsinstitut Oberwolfach, November 7, 2023.

D. Peschka, Variational approach for wetting flows on solid and soft substrates (online talk), Colloquium of the SPP 2171 (Online Event), Westfälische WilhelmsUniversität Münster, January 28, 2022.

M. Thomas, Firstorder formulation for dynamic phasefield fracture in viscoelastic materials, PHAse field MEthods in applied sciences (PHAME 2022), May 23  27, 2022, Istituto Nazionale di Alta Matematica, Rome, Italy, May 25, 2022.

M. Thomas, Firstorder formulation for dynamic phasefield fracture in viscoelastic materials, Beyond Elasticity: Advances and Research Challenges, May 16  20, 2022, Centre International de Rencontres Mathématiques, Marseille, France, May 16, 2022.

M. Thomas, Firstorder formulation for dynamic phasefield fracture in viscoelastic materials, Jahrestreffen des SPP 2256, September 28  30, 2022, Universität Regensburg, September 30, 2022.

A. Alphonse, Directional differentiability and optimal control for elliptic quasivariational inequalities (online talk), Workshop ``Challenges in Optimization with Complex PDESystems'' (Hybrid Workshop), February 14  20, 2021, Mathematisches Forschungsinstitut Oberwolfach, February 17, 2021.

A. Alphonse, Directional differentiability and optimal control for elliptic quasivariational inequalities (online talk), Meeting of the Scientific Advisory Board of WIAS, WIAS Berlin, March 12, 2021.

A. Alphonse, Some aspects of sensitivity analysis and optimal control for elliptic QVIs (coauthors: Michael Hintermüller and Carlos Rautenberg, online talk), 91th Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Session DFGPP 1962 Nonsmooth and Complementaritybased Distributed Parameter Systems, March 15  19, 2021, Universität Kassel, March 16, 2021.

A. Alphonse, Some aspects of sensitivity analysis and optimal control for elliptic QVIs (online talk), Annual Meeting of the DFG SPP 1962 (Virtual Conference), March 24  25, 2021, WIAS Berlin, March 25, 2021.

X. Liu, Second law of thermodynamics and bounded entropy solutions in the compressible NavierStokes system, Summer School 2021 ``Wave Phenomena: Analysis and Numerics'' (Hybrid Event), September 27  30, 2021.

D. Peschka, Dynamic contact angles and their numerical discretization for gradient systems (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (Online Event), September 13  17, 2021, WIAS Berlin, September 13, 2021.

D. Peschka, Mathematical modeling and simulation of substrateflow interaction using generalized gradient flows, Conference ``Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates'' of the SPP 2171, Freiburg, November 8  10, 2021.

D. Peschka, Mathematical modeling and simulation of substrateflow interaction using generalized gradient flows, SPP 2171 Conference ``Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates'', November 8  10, 2021, AlbertLudwigsUniversität Freiburg, November 10, 2021.

D. Peschka, Selfsimilar spreading with dynamic contact angles (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S11 ``Interfacial Flows'', March 15  19, 2021, Universität Kassel, March 17, 2021.

D. Peschka, Thinfilm problems with dynamic contact angle (online talk), 8th European Congress of Mathematics (8ECM), Minisymposium ID 43 ``Higherorder Evolution Equations'' (Online Event), June 20  26, 2021, Portoroč, Slovenia, June 23, 2021.

C. Geiersbach, Stochastic approximation for optimization in shape spaces, 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (FBP 2021, Online Event), Minisymposium ``UQ in Free Boundary Problems'', September 13  17, 2021, WIAS, Berlin, September 14, 2021.

M. Thomas, Convergence analysis for fully discretized damage and phasefield fracture models (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (FBP 2021, Online Event), Minisymposium ``Phase Field Models'', September 13  17, 2021, WIAS, Berlin, September 14, 2021.

M. Hintermüller, Optimal control of quasivariational inequalities (online talk), SIAM Conference on Optimization (OP21) (Online Event), Minisymposium MS93 ``Nonsmooth Problems and Methods in Largescale Optimization'', July 20  23, 2021, July 23, 2021.

X. Liu, Second law of thermodynamics and bounded entropy solutions in the compressible NavierStokes system (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (Online Event), September 13  17, 2021, WIAS Berlin, September 17, 2021.

D. Peschka, Mathematical modeling and simulation of flows and the interaction with a substrate using energetic variational methods, CRC 1194 ``Interaction between Transport and Wetting Processes'', Technische Universität Darmstadt, January 22, 2020.

D. Peschka, Variational modeling of bulk and interface effects in fluid dynamics, SPP 2171 Advanced School ``Introduction to Wetting Dynamics'', February 17  21, 2020, Westfälische WilhelmsUniversität Münster, February 18, 2020.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, MATH+ Day 2020 (Online Event), Berlin, November 6, 2020.

M. Thomas, Modeling and analysis of flows of concentrated suspensions (online talk), Colloquium of the RTG 2339 ``Interfaces, Complex Structures, and Singular Limits'' (Online Event), Universität Regensburg, July 10, 2020.

M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, January 30, 2020.

M. Thomas, Thermodynamical modelling via energy and entropy functionals (online talks), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin.

M. Thomas, WeierstraßGruppe "VolumenGrenzschichtProzesse", Sitzung des Wissenschaftlichen Beirats, WIAS Berlin, September 18, 2020.

B. Wagner, Pattern formation in dewetting films (online talk), Workshop ``Mathematical Modeling and Scientific Computing: Focus on Complex Processes and Systems'' (Online Event), November 19  20, 2020, Technische Universität München, November 19, 2020.

B. Wagner, Phasefield models of the lithiation/delithiation cycle of thinfilm electrodes (online talk), Oxford Battery Modelling Symposium (Online Event), March 16  17, 2020, University of Oxford, UK, March 16, 2020.

J.A. Brüggemann, Elliptic obstacletype quasivariational inequalities (QVIs) with volume constraints motivated by a contact problem in biomedicine, ICCOPT 2019  Sixth International Conference on Continuous Optimization, Berlin, August 5  8, 2019.

J.A. Brüggemann, Solution methods for a class of obstacletype quasi variational inequalities with volume constraints, ICCOPT 2019  Sixth International Conference on Continuous Optimization, Session ``QuasiVariational Inequalities and Generalized Nash Equilibrium Problems (Part II)'', August 5  8, 2019, Berlin, August 7, 2019.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, Visit of the Scientific Advisory Board of MATH+, November 11, 2019.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, 1st MATH+ Day, Berlin, December 13, 2019.

D. Peschka, Dynamic contact angles via generalized gradient flows, Modelling of Thin Liquid Films  Asymptotic Approach vs. Gradient Dynamics, April 28  May 3, 2019, Banff International Research Station for Mathematical Information and Discovery, Canada, April 30, 2019.

D. Peschka, Dynamic contact angles via gradient flows, 694. WEHeraeusSeminar ``Wetting on Soft or Microstructured Surfaces'', Bad Honnef, April 10  13, 2019.

D. Peschka, Gradient formulations with flow maps  Mathematical and numerical approaches to free boundary problems, Kolloquium des Graduiertenkollegs 2339 ``Interfaces, Complex Structures, and Singular Limits'', Universität Regensburg, May 24, 2019.

D. Peschka, Mathematical modeling and simulation of substrateflow interaction using generalized gradient flow, Begutachtungskolloquium für die Anträge des SPP 2171 ``Dynamische Benetzung flexibler, adaptiver und schaltbarer Oberflächen'', Mainz, February 7  8, 2019.

D. Peschka, Mathematical modeling of fluid flows using gradient systems, Seminar in PDE and Applications, Delft University of Technology, Netherlands, May 28, 2019.

D. Peschka, Steering pattern formation of viscous flows, DMVJahrestagung 2019, Sektion ``Differentialgleichungen und Anwendungen'', September 23  26, 2019, KIT  Karlsruher Institut für Technologie, September 23, 2019.

M. Thomas, Coupling of rateindependent and ratedependent systems, MURPHYSHSFS 2019 Summer School on MultiRate Processes, SlowFast Systems and Hysteresis, June 17  19, 2019, Politecnico di Torino, Turin, Italy.

M. Thomas, Coupling of rateindependent and ratedependent systems with application to delamination processes in solids, Mathematics for Mechanics, October 29  November 1, 2019, Czech Academy of Sciences, Prague, Czech Republic, October 31, 2019.

M. Thomas, Coupling of rateindependent and ratedependent systems with application to delamination processes in solids, Seminar ``Applied and Computational Analysis'', University of Cambridge, UK, October 10, 2019.

M. Thomas, GENERIC structures with bulkinterface interaction, SFB 910 Symposium ``Energy Based Modeling, Simulation and Control'', October 25, 2019, Technische Universität Berlin, October 25, 2019.

M. Thomas, Gradient structures for flows of concentrated suspensions, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS ME775 ``Recent Advances in Understanding Suspensions and Granular Media Flow'', July 15  19, 2019, Valencia, Spain, July 17, 2019.

M. Thomas, Rateindependent evolution of sets and application to fracture processes, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18  22, 2019, Technische Universität Wien, Austria, February 20, 2019.

B. Wagner, Free boundary problems of active and driven hydrogels, PIMSGermany Workshop on Modelling, Analysis and Numerical Analysis of PDEs for Applications, June 24  26, 2019, Universität Heidelberg, Interdisciplinary Center for Scientific Computing and BIOQUANT Center, June 24, 2019.

B. Wagner, Free boundary problems of active and driven hydrogels, EUROMECH 604, Fluid and Solid Mechanics for Issue Engineering, September 23  25, 2019, University of Oxford, Mathematical Institute, UK, September 24, 2019.

M. Hintermüller, Lecture Series: Optimal control of nonsmooth structures, Thematic Programme ``Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions'', February 4  7, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Colloquium of the Mathematical Institute, University of Oxford, UK, June 7, 2019.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland, December 3, 2019.

J.A. Brüggemann, Pathfollowing methods for a class of elliptic obstacletype quasivariational problems with integral constraints, 23rd International Symposium on Mathematical Programming (ISMP2018), Session 370 ``Variational Analysis 4'', July 1  6, 2018, Bordeaux, France, July 2, 2018.

M. Thomas, D. Peschka, B. Wagner, V. Mehrmann, M. Rosenau, Modeling and analysis of suspension flows, MATH+ Center Days 2018, October 31  November 2, 2018, ZuseInstitut Berlin (ZIB), Berlin, October 31, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section DFG Priority Programmes PP1748 ``Reliable Simulation Techniques in Solid Mechanics. Development of Nonstandard Discretization Methods, Mechanical and Mathematical Analysis'', March 19  23, 2018, Technische Universität München, March 20, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18  22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 18, 2018.

M. Thomas, Analysis and simulation for a phasefield fracture model at finite strains based on modified invariants, Analysis Seminar, University of Brescia, Department of Mathematics, Italy, May 10, 2018.

M. Thomas, Gradient structures for flows of concentrated suspensions, The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 18 ``Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations and Related Fields'', July 5  9, 2018, National Taiwan University, Taipeh, Taiwan, Province Of China, July 7, 2018.

M. Hintermüller, Nonsmooth structures in PDE constrained optimization, Mathematisches Kolloquium, Universität Bielefeld, Fakultät für Mathematik, June 7, 2018.

C.N. Rautenberg, Dissipative and nondissipative evolutionary quasivariational inequalities with derivative constraints, Joint Meeting of the Italian Mathematical Union, the Italian Society of Industrial and Applied Mathematics and the Polish Mathematical Society, Session 14 ``Nonlinear Variational Methods with Applications'', September 17  20, 2018, Wroclaw, Poland, September 19, 2018.

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.

D. Peschka, Mathematical and numerical approaches to moving contact lines, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, December 6, 2017.

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.

D. Peschka, Variational structure of fluid motion with contact lines in thinfilm models, Kolloquium Angewandte Mathematik, Universität der Bundeswehr, München, May 31, 2017.

M. Hintermüller, Adaptive finite element solvers for MPECs in function space, SIAM Conference on Optimization, Minisymposium MS122 ``Recent Trends in PDEConstrained Optimization'', May 22  25, 2017, Vancouver, British Columbia, Canada, May 25, 2017.

M. Hintermüller, Generalized Nash games with partial differential equations, Kolloquium Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen, Technische Universität Berlin, June 20, 2017.

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

M. Hintermüller, Nonsmooth structures in PDE constrained optimization, Optimization Seminar, Chinese Academy of Sciences, State Key Laboratory of Scientific and Engineering Computing, Beijing, China, June 6, 2017.

M. Hintermüller, Optimal control of multiphase fluids and droplets, Kolloquium, FriedrichAlexanderUniversität ErlangenNürnberg, Department Mathematik, Erlangen, May 2, 2017.

M. Hintermüller, Optimal control of multiphase fluids based on non smooth models, 14th International Conference on Free Boundary Problems: Theory and Applications, Theme Session 8 ``Optimization and Control of Interfaces'', July 9  14, 2017, Shanghai Jiao Tong University, China, July 10, 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.

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, Multiphase 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, CeNoSKolloquium, Westfälische WilhelmsUniversität 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 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 and Applications, Spring School on Nonlinear PDEs and Related Problems, January 15  19, 2016, African Institute for Mathematical Sciences (AIMS), Mbour, Senegal.

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, JustusLiebigUniversität 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, 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.

M.H. Farshbaf Shaker, A phase field approach for optimal boundary control of damage processes in twodimensional 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, WIASPGMO 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 PDEconstrained 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 Infinitedimensional 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 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. Heida, Modeling of fluid interfaces, Jahrestagung der Deutschen MathematikerVereinigung, 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 Thinfilm 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, Thinfilm equations with free boundaries, Jahrestagung der Deutschen MathematikerVereinigung, 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?, PerlenKolloquium, Universität Basel, Fachbereich Mathematik, Switzerland, May 22, 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 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.

M.H. Farshbaf Shaker, A deep quench approach to the optimal control of an AllenCahn 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, SADCOWIAS 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 CahnHilliard 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, Kickoff Symposium Mathematics: Official Opening of the Felix Klein Building, FriedrichAlexanderUniversität ErlangenNürnberg, 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 JacquesLouis Lions, Paris, France, September 30, 2009.

W. Dreyer, Phase transitions during hydrogen storage and in lithiumion 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.
Preprints im Fremdverlag

L. Caffarelli, X. RosOton, J. Serra, Obstacle problems for integrodifferential 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 integrodifferential 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 integrodierential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicitytype for mula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integrodierential operators: we establish similar regularity results for obstacle problems with convex operators. 
X. RosOton, 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 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.