Functional analysis and operator theory

H. Neidhardt, A. Stephan, V.A. Zagrebnov, Chapter 13: Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces, in: Analysis and Operator Theory, Th.M. Rassias , V.A. Zagrebnov , eds., 146 of Springer Optimization and Its Applications, Springer, Cham, 2019, pp. 271--299, (Chapter Published), DOI 10.1007/978-3-030-12661-2_13 .

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).

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. Exner, W. König, H. Neidhardt, eds., Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, World Scientific Publishing, Singapore, 2015, xii+383 pages, (Collection Published).

V. Maz'ya, G. Schmidt, Approximate Approximations, 141 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2007, 349 pages, (Monograph Published).

P.-É. Druet, A. Jüngel, Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient, SIAM Journal on Mathematical Analysis, 52 (2020), pp. 2179--2197, DOI 10.1137/19M1301473 .
Abstract
The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.

H. Neidhardt, A. Stephan, V.A. Zagrebnov, Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems, Publications of the Research Institute for Mathematical Sciences, 56 (2020), pp. 83--135, DOI 10.4171/PRIMS/56-1-5 .
Abstract
In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(J,X), consisting of X-valued functions on the time-interval J. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(J,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.

K. Disser, J. Rehberg, The 3D transient semiconductor equations with gradient-dependent and interfacial recombination, Mathematical Models & Methods in Applied Sciences, 29 (2019), pp. 1819--1851, DOI 10.1142/S0218202519500350 .
Abstract
We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.

A. Bacho, E. Emmrich, A. Mielke, An existence result and evolutionary Gamma-convergence for perturbed gradient systems, Journal of Evolution Equations, 19 (2019), pp. 479--522, DOI 10.1007/s00028-019-00484-x .
Abstract
We consider the initial-value problem for the perturbed gradient flows, where a differential inclusion is formulated in terms of a subdifferential of an energy functional, a subdifferential of a dissipation potential and a more general perturbation, which is assumed to be continuous and to satisfy a suitable growth condition. Under additional assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique.

V. Laschos, A. Mielke, Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures, Journal of Functional Analysis, 276 (2019), pp. 3529--3576, DOI 10.1016/j.jfa.2018.12.013 .
Abstract
By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and reparametrization of the geodesics, local-angle condition and some partial K-semiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows.

A.F.M. TER Elst, J. Rehberg, Consistent operator semigroups and their interpolation, Journal of Operator Theory, 82 (2019), pp. 3--21, DOI 10.7900/jot.2017nov15.2233 .
Abstract
Under a mild regularity condition we prove that the generator of the interpolation of two C₀-semigroups is the interpolation of the two generators.

P.-É. Druet, Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact, Mathematical Methods in the Applied Sciences, 41 (2018), pp. 6457--6479, DOI 10.1002/mma.5170 .
Abstract
We investigate the regularity of the weak solution to elliptic transmission problems that involve several materials intersecting at a closed interior line of contact. We prove that local weak solutions possess second order generalized derivatives up to the contact line, mainly exploiting their higher regularity in the direction tangential to the line. Moreover we are thus able to characterize the higher regularity of the gradient and the Hoelder exponent by means of explicit estimates known in the literature for two dimensional problems. They show that strong regularity properties, for instance the integrability of the gradient to a power larger than the space dimension d =3, are to expect if the oscillations of the diffusion coefficient are moderate (that is for far larger a range than what a theory of small perturbations would allow), or if the number of involved materials does not exceed three.

M. Heida, R.I.A. Patterson, D.R.M. Renger, Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space, Journal of Evolution Equations, 19 (2019), pp. 111--152 (published online on 14.09.2018), DOI 10.1007/s00028-018-0471-1 .
Abstract
We study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin-Lions theorem. We finally provide some useful applications to stochastic processes.

K. Disser, M. Liero, J. Zinsl, On the evolutionary Gamma-convergence of gradient systems modeling slow and fast chemical reactions, Nonlinearity, 31 (2018), pp. 3689--3706, DOI 10.1088/1361-6544/aac353 .
Abstract
We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Γ-convergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudo-metric.

D. Horstmann, J. Rehberg, H. Meinlschmidt, The full Keller--Segel model is well-posed on fairly general domains, Nonlinearity, 31 (2018), pp. 1560--1592, DOI 10.1088/1361-6544/aaa2e1 .
Abstract
In this paper we prove the well-posedness of the full Keller-Segel system, a quasilinear strongly coupled reaction-crossdiffusion system, in the spirit that it always admits a unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. Apparently, there exists no comparable existence result for the full Keller-Segel system up to now. The proof is carried out for general source terms and is based on recent nontrivial elliptic and parabolic regularity results which hold true even on fairly general spatial domains, combined with an abstract solution theorem for nonlocal quasilinear equations by Amann.

M. Liero, S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, NoDEA. Nonlinear Differential Equations and Applications, 25 (2018), pp. 6/1--6/31, DOI 10.1007/s00030-018-0495-9 .
Abstract
In this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

K. Disser, A.F.M. TER Elst, J. Rehberg, On maximal parabolic regularity for non-autonomous parabolic operators, Journal of Differential Equations, 262 (2017), pp. 2039--2072.
Abstract
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic L^r-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

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

M. Liero, A. Mielke, G. Savaré, Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures, Inventiones mathematicae, 211 (2018), pp. 969--1117 (published online on 14.12.2017), DOI 10.1007/s00222-017-0759-8 .
Abstract
We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1--35, DOI 10.3934/dcdss.2017001 .
Abstract
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

M. Mittnenzweig, A. Mielke, An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models, Journal of Statistical Physics, 167 (2017), pp. 205--233, DOI 10.1007/s10955-017-1756-4 .
Abstract
We show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems.

H. Neidhardt, A. Stephan, V.A. Zagrebnov, On convergence rate estimates for approximations of solution operators of linear non-autonomous evolution equations, Nanosystems: Physics, Chemistry, Mathematics, 8 (2017), pp. 202--215, DOI 10.17586/2220-8054-2017-8-2-202-215 .
Abstract
We improve some recent estimates of the rate of convergence for product approximations of solution operators for linear non-autonomous Cauchy problem. The Trotter product formula approximation is proved to converge to the solution operator in the operator-norm. We estimate the rate of convergence of this approximation. The result is applied to diffusion equation perturbed by a time-dependent potential.

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. 1--30, DOI 10.4171/IFB/375 .

J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering matrices and Dirichlet-to-Neumann maps, Journal of Functional Analysis, 273 (2017), pp. 1970--2025, DOI 10.1016/j.jfa.2017.06.001 .
Abstract
A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh?Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.

P. Exner, A.S. Kostenko, M.M. Malamud, H. Neidhardt, Infinite quantum graphs, Rossiiskaya Akademiya Nauk. Doklady Akademii Nauk, 472 (2017), pp. 253--258, DOI 10.1134/S1064562417010136 .
Abstract
Infinite quantum graphs with ?-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.

M.M. Malamud, H. Neidhardt, H. Peller, A trace formula for functions of contractions and analytic operator Lipschitz functions, Comptes Rendus Mathematique. Academie des Sciences. Paris, 355 (2017), pp. 806--811, DOI 10.1016/j.crma.2017.06.003 .
Abstract
In this note, we study the problem of evaluating the trace of $f(T) - F(R)$, where $T$ and $R$ are contractions on a Hilbert space with trace class difference, i.e. $T-R in mathbf S_1$, and $f$ is a function analytic in the unit disk $mathbb D$. It is well known that if $f$ is an operator Lipschitz function analytic in $mathbb D$, then $f(T) - f(R) in mathbf S_1$. The main result of the note says that there exists a function $xi$ (a spectral shift function) on the unit circle $mathbb T$ of class $L^1(mathbb T)$ such that the following trace formula holds: $tr(f(T) - f(R))= int_mathbbT f'(zeta)xi(zeta)dzeta$, whenever $T$ and $R$ are contractions with $T-R in mathbf S_1$, and $f$ is an operator Lipschitz function analytic in $mathbb D$.

H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions. Part 1: Existence of optimal solutions, SIAM Journal on Control and Optimization, 55 (2017), pp. 2876--2904, DOI 10.1137/16M1072644 .
Abstract
This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

H. Meinlschmidt, Ch. Meyer, J. Rehberg, Optimal control of the thermistor problem in three spatial dimensions. Part 2: Optimality conditions, SIAM Journal on Control and Optimization, 55 (2017), pp. 2368--2392, DOI 10.1137/16M1072656 .
Abstract
This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

A. VAN Rooij, W. van Zuijlen, Bochner integrals in ordered vector spaces, Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity, 21 (2017), pp. 1089--1113.

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

M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, Analytical aspects of spatially adapted total variation regularisation, Journal of Mathematical Analysis and Applications, 454 (2017), pp. 891--935, DOI 10.1016/j.jmaa.2017.05.025 .
Abstract
In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions.

M. Hintermüller, C.N. Rautenberg, S. Rösel, Density of convex intersections and applications, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 473 (2017), pp. 20160919/1--20160919/28, DOI 10.1098/rspa.2016.0919 .
Abstract
In this paper we address density properties of intersections of convex sets in several function spaces. Using the concept of Gamma-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite element discretizations of sets associated to convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.

M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017), pp. 1--35.
Abstract
A class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type.

M. Hintermüller, C.N. Rautenberg, Optimal selection of the regularization function in a weighted total variation model. Part I: Modeling and theory, Journal of Mathematical Imaging and Vision, 59 (2017), pp. 498--514.
Abstract
Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.

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

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

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

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

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

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

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

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

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

M.M. Malamud, H. Neidhardt, Trace formulas for additive and non-additive perturbations, Advances in Mathematics, 274 (2015), pp. 736--832.
Abstract
Trace formulas for pairs of self-adjoint, maximal dissipative and other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a resolvent comparable pair H', H of maximal dissipative operators is proved. We also investigate the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H and H'=H+V are maximal dissipative and V is of trace class, we prove the existence of a summable complex-valued spectral shift function. We also obtain trace formulas for a pair {A, A*} assuming only that A and A* are resolvent comparable. In this case the determinant of a characteristic function of A is involved in the trace formula.

In the case of singular perturbations we apply the technique of boundary triplets. It allows to express the spectral shift function of a pair of extensions in terms of abstract Weyl function and boundary operator.

We improve and generalize certain classical results of M.G. Krein for pairs of self-adjoint and dissipative operators, the results of A. Rybkin for such pairs, as well as the results of V. Adamyan, B. Pavlov, and M. Krein for pairs {A, A*} with a maximal dissipative operator A.

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

R. Ferreira, C. Kreisbeck, A.M. Ribeiro, Characterization of polynomials and higher-order Sobolev spaces in terms of nonlocal functionals involving difference quotients, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, (available online on Oct. 6, 2014), DOI 10.1016/j.na.2014.09.007 .
Abstract
The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a kth-order Sobolev space. One of the main theorems is a new characterization of W^k,p(Ω), k∈ ℕ and p ∈ (1, +∞), for arbitrary open sets Ω ⊂ ℝⁿ. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis' overview article [Russ. Math. Surv. 57 (2002), pp. 693-708] to the higher-order case, and extend the work by Borghol [Asymptotic Anal. 51 (2007), pp. 303-318] to a more general setting.

H. Cornean, H. Neidhardt, L. Wilhelm, V. Zagrebnov, The Cayley transform applied to non-interacting quantum transport, Journal of Functional Analysis, 266 (2014), pp. 1421--1475.
Abstract
We extend the Landauer-Büttiker formalism in order to accommodate both unitary and self-adjoint operators which are not bounded from below. We also prove that the pure point and singular continuous subspaces of the decoupled Hamiltonian do not contribute to the steady current. One of the physical applications is a stationary charge current formula for a system with four pseudo-relativistic semi-infinite leads and with an inner sample which is described by a Schrödinger operator defined on a bounded interval with dissipative boundary conditions. Another application is a current formula for electrons described by a one dimensional Dirac operator; here the system consists of two semi-infinite leads coupled through a point interaction at zero.

M. Malamud, H. Neidhardt, Perturbation determinants for singular perturbations, Russian Journal of Mathematical Physics, 21 (2014), pp. 55--98.
Abstract
For proper extensions of a densely defined closed symmetric operator with trace class resolvent difference the perturbation determinant is studied in the framework of boundary triplet approach to extension theory.

S. Albeverio, A. Kostenko, M. Malamud, H. Neidhardt, Spherical Schrödinger operators with $delta$-type interactions, Journal of Mathematical Physics, 54 (2013), pp. 052103/1--052103/24.
Abstract
We investigate spectral properties of spherical Schrödinger operators (also known as Bessel operators) with $delta$-point interactions concentrated on a discrete set. We obtain necessary and sufficient conditions for these Hamiltonians to be self-adjoint, lower-semibounded and also we investigate their spectra.We also extend the classical Bargmann estimate to such Hamiltonians. In certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities. We apply our results to Schr¨odinger operators in $L^2(mathbbR^n)$ with a singular interaction supported by an infinite family of concentric spheres.

A.A. Boitsev, H. Neidhardt, I. Popov, Weyl function for sum of operator tensor products, Nanosystems: Physics, Chemistry, Mathematics, 4 (2013), pp. 747--757.

M. Malamud, H. Neidhardt, Sturm--Liouville boundary value problems with operator potentials and unitary equivalence, Journal of Differential Equations, 252 (2012), pp. 5875--5922.
Abstract
Consider the minimal Sturm-Liouville operator $A = A_rm min$ generated by the differential expression $cA := -fracd^2dt^2 + T$ in the Hilbert space $L^2(R_+,cH)$ where $T = T^*ge 0$ in $cH$. We investigate the absolutely continuous parts of different self-adjoint realizations of $cA$. In particular, we show that Dirichlet and Neumann realizations, $A^D$ and $A^N$, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if $infsigma_ess(T) = infgs(T) ge 0$, then the part $wt A^acE_wt A(gs(A^D))$ of any self-adjoint realization $wt A$ of $cA$ is unitarily equivalent to $A^D$. In addition, we prove that the absolutely continuous part $wt A^ac$ of any realization $wt A$ is unitarily equivalent to $A^D$ provided that the resolvent difference $(wt A - i)^-1- (A^D - i)^-1$ is compact. The abstract results are applied to elliptic differential expression in the half-space.

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

P. Exner, H. Neidhardt, V. Zagrebnov, Remarks on the Trotter--Kato product formula for unitary groups, Integral Equations and Operator Theory, 69 (2011), pp. 451--478.
Abstract
Let $A$ and $B$ be non-negative self-adjoint operators in a separable Hilbert space such that its form sum $C$ is densely defined. It is shown that the Trotter product formula holds for imaginary times in the $L^2$-norm. The result remains true for the Trotter-Kato product formula for so-called holomorphic Kato functions; we also derive a canonical representation for any function of this class.

M.M. Malamud, H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, Journal of Functional Analysis, 260 (2011), pp. 613--638.
Abstract
The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $gotH$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $widetilde A = widetilde A^*$ and $A_0$ are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function $M(cdot)$ of a pair $A,A_0$ admits bounded limits $M(t) := wlim_yto+0M(t+iy)$ for a.e. $t in mathbbR$. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.

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

M.M. Malamud, H. Neidhardt, On Kato--Rosenblum and Weyl--von Neumann theorems, Rossiiskaya Akademiya Nauk. Doklady Akademii Nauk, 432 (2010), pp. 161--166.

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

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

P.N. Racec, R. Racec, H. Neidhardt, Evanescent channels and scattering in cylindrical nanowire heterostructures, Phys. Rev. B., 79 (2009), pp. 155305/1--155305/14.
Abstract
We investigate the scattering phenomena produced by a general finite range non-separable potential in a multi-channel two-probe cylindrical nanowire heterostructure. The multi-channel current scattering matrix is efficiently computed using the R-matrix formalism extended for cylindrical coordinates. Considering the contribution of the evanescent channels to the scattering matrix, we are able to put in evidence the specific dips in the tunneling coefficient in the case of an attractive potential. The cylindrical symmetry cancels the ”selection rules” known for Cartesian coordinates. If the attractive potential is superposed over a non-uniform potential along the nanowire, then resonant transmission peaks appear. We can characterize them quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances (dips and peaks). Our formalism is applied to a variety of model systems like a quantum dot, a core/shell quantum ring or a double barrier, embedded into the nano-cylinder.

H.D. Cornean, H. Neidhardt, V.A. Zagrebnov, The effect of time-dependent coupling on non-equilibrium steady states, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 10 (2009), pp. 61--93.
Abstract
Consider (for simplicity) two one-dimensional semi-infinite leads coupled to a quantum well via time dependent point interactions. In the remote past the system is decoupled, and each of its components is at thermal equilibrium. In the remote future the system is fully coupled. We define and compute the non equilibrium steady state (NESS) generated by this evolution. We show that when restricted to the subspace of absolute continuity of the fully coupled system, the state does not depend at all on the switching. Moreover, we show that the stationary charge current has the same invariant property, and derive the Landau-Lifschitz and Landauer-Büttiker formulas.

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

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

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

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

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

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

J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering matrices and Weyl functions, Proceedings of the London Mathematical Society. Third Series, 97 (2008), pp. 568--598.
Abstract
For a scattering system consisting of two selfadjoint extensions of a symmetric operator A with finite deficiency indices, the scattering matrix and the spectral shift function are calculated in terms of the Weyl function associated with the boundary triplet for A* and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar- and matrix-valued potentials, to Dirac operators and to Schroedinger operators with point interactions.

J. Behrndt, H. Neidhardt, R. Racec, P.N. Racec, U. Wulf, On Eisenbud's and Wigner's R-matrix: A general approach, Journal of Differential Equations, 244 (2008), pp. 2545--2577.
Abstract
The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis.

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

R. Haller-Dintelmann, M. Hieber, J. Rehberg, Irreducibility and mixed boundary conditions, Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity, 12 (2008), pp. 83--91.

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

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

J.A.C. Martins, M.D.P. Monteiro Marques, A. Petrov, On the stability of elastic-plastic systems with hardening, Journal of Mathematical Analysis and Applications, 343 (2008), pp. 1007--1021.
Abstract
This paper discusses the stability of quasi-static paths for a continuous elastic-plastic system with hardening in a one-dimensional (bar) domain. Mathematical formulations, as well as existence and uniqueness results for dynamic and quasi-static problems involving elastic-plastic systems with linear kinematic hardening are recalled in the paper. The concept of stability of quasi-static paths used here is essentially a continuity property of the system dynamic solutions relatively to the quasi-static ones, when (as in Lyapunov stability) the size of initial perturbations is decreased and the rate of application of the forces (which plays the role of the small parameter in singular perturbation problems) is also decreased to zero. The stability of the quasi-static paths of these elastic-plastic systems is the main result proved in the paper.

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

J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering theory for open quantum systems with finite rank coupling, Mathematical Physics, Analysis and Geometry. An International Journal Devoted to the Theory and Applications of Analysis and Geometry to Physics, 10 (2007), pp. 313--358.
Abstract
Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator $A_D$ in a Hilbert space $sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation $widetilde K$ of $A_D$ can be regarded as the Hamiltonian of a closed system which contains the open system $[A_D,sH]$, but since $widetilde K$ is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family $[A(mu)]$ of maximal dissipative operators depending on energy $mu$, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.

P. Exner, T. Ichinose, H. Neidhardt, V. Zagrebnov, Zeno product formula revisited, Integral Equations and Operator Theory, 57 (2007), pp. 67--81.

J.A.C. Martins, M.D.P. Monteiro, A. Petrov, On the stability of quasi-static paths for finite dimensional elastic-plastic systems with hardening, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 87 (2007), pp. 303--313.

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

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

H. Neidhardt, J. Rehberg, Scattering matrix, phase shift, spectral shift and trace formula for one-dimensional Schrödinger-type operators, Integral Equations and Operator Theory, 58 (2007), pp. 407--431.
Abstract
The paper is devoted to Schroedinger operators on bounded intervals of the real axis with dissipative boundary conditions. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed, in particular, trace formula and Birman-Krein formula are verified directly. The results are used for dissipative Schroedinger-Poisson systems.

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

H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Convexity of trace functionals and Schrödinger operators, Journal of Functional Analysis, 234 (2006), pp. 45--69.

M. Baro, N. Ben Abdallah, P. Degond, A. El Ayyadi, A 1D coupled Schrödinger drift-diffusion model including collisions, Journal of Computational Physics, 203 (2005), pp. 129-153.

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

M. Baro, M. Demuth, E. Giere, Stable continuous spectra for differential operators of arbitrary order, Analysis and Applications, 3 (2005), pp. 223-250.

TH. Koprucki, M. Baro, U. Bandelow, Th. Tien, F. Weik, J.W. Tomm, M. Grau, M.-Ch. Amann, Electronic structure and optoelectronic properties of strained InAsSb/GaSb multiple quantum wells, Applied Physics Letters, 87 (2005), pp. 181911/1--181911/3.

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

S. Albeverio, J.F. Brasche, M.M. Malamud, H. Neidhardt, Inverse spectral theory for symmetric operators with several gaps: Scalar-type Weyl functions, Journal of Functional Analysis, 228 (2005), pp. 144--188.

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

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

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

T. Ichinose, H. Neidhardt, V.A. Zagrebnov, Trotter--Kato product formula and fractional powers of self-adjoint generators, Journal of Functional Analysis, 207 (2004), pp. 33--57.

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

M. Baro, H. Neidhardt, Dissipative Schrödinger-type operator as a model for generation and recombination, Journal of Mathematical Physics, 44 (2003), pp. 2373--2401.

G. Bruckner, S.V. Pereverzev, Self-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 19 (2003), pp. 147--156.

H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Macroscopic current induced boundary conditions for Schrödinger-type operators, Integral Equations and Operator Theory, 45 (2003), pp. 39--63.

H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, On 1-dimensional dissipative Schrödinger-type operators, their dilations and eigenfunction expansions, Mathematische Nachrichten, 252 (2003), pp. 51--69.

J.F. Brasche, M. Malamud, H. Neidhardt, Weyl function and spectral properties of self-adjoint extensions, Integral Equations and Operator Theory, 43 (2002), pp. 264--289.

V. Cachia, H. Neidhardt, V.A. Zagrebnov, Comments on the Trotter product formula error-bound estimates for nonself-adjoint semigroups, Integral Equations and Operator Theory, 42 (2002), pp. 425--448.

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

H.-Chr. Kaiser, H. Neidhardt, J. Rehberg, Density and current of a dissipative Schrödinger operator, Journal of Mathematical Physics, 43 (2002), pp. 5325--5350.

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

V. Cachia, H. Neidhardt, V.A. Zagrebnov, Accretive perturbations and error estimates for the Trotter product formula, Integral Equations and Operator Theory, 39 (2001), pp. 396--412.

P. Exner, H. Neidhardt, V.A. Zagrebnov, Potential approximation to $delta'$: An inverse Klauder phenomenon with norm-resolvent convergence, Communications in Mathematical Physics, 224 (2001), pp. 593--612.

U. Bandelow, H.-Chr. Kaiser, Th. Koprucki, J. Rehberg, Spectral properties of $k cdot p$ Schrödinger operators in one space dimension, Numerical Functional Analysis and Optimization. An International Journal, 21 (2000), pp. 379--409.

V.M. Adamyan, H. Neidhardt, On the absolutely continuous subspace for non-selfadjoint operators, , 210 (2000), pp. 5--42.

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

M. Bulíček, A. Glitzky, M. Liero, Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions, in: PDE 2015: Theory and Applications of Partial Differential Equations, H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., 10 of Discrete and Continuous Dynamical Systems, Series S, no. 4, American Institute of Mathematical Sciences, Springfield, 2017, pp. 697--713.
Abstract
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L¹ term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

Y. Granovskyi, M.M. Malamud, H. Neidhardt, A. Posilicano, To the spectral theory of vector-valued Sturm--Liouville operators with summable potentials and point interactions, in: Functional Analysis and Operator Theory for Quantum Physics, J. Dittrich, H. Kovarik, A. Laptev, eds., EMS Series of Congress Reports, EMS Publishing House, 2017, pp. 271--313, DOI 10.4171/175-1/15 .

V. Lotoreichik, H. Neidhardt, I.Y. Popov, Point contacts and boundary triples, in: Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 283--293.
Abstract
We suggest an abstract approach for point contact problems in the framework of boundary triples. Using this approach we obtain the perturbation series for a simple eigenvalue in the discrete spectrum of the model self-adjoint extension with weak point coupling.

M. Malamud, H. Neidhardt, Trace formulas for singular and additive non-selfadjoint perturbations, in: Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, P. Exner, W. König, H. Neidhardt, eds., World Scientific Publishing, Singapore, 2015, pp. 295--303.

J. Behrndt, M.M. Malamud, H. Neidhardt, Finite rank perturbations, scattering matrices and inverse problems, in: Operator Theory in Krein Spaces and Spectral Analysis, J. Behrndt, K.-H. Förster, C. Trunk, H. Winkler, eds., 198 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2009, pp. 61--85.
Abstract
In this paper the scattering matrix of a scattering system consisting of two selfadjoint operators with finite dimensional resolvent difference is expressed in terms of a matrix Nevanlinna function. The problem is embedded into an extension theoretic framework and the theory of boundary triplets and associated Weyl functions for (in general nondensely defined) symmetric operators is applied. The representation results are extended to dissipative scattering systems and an explicit solution of an inverse scattering problem for the Lax-Phillips scattering matrix is presented.

J. Behrndt, M. Malamud, H. Neidhardt, Trace formula for dissipative and coupled scattering systems, in: Spectral Theory in Inner Product Spaces and Applications, J. Behrndt, K.-H. Förster, H. Langer, C. Trunk, eds., 188 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2008, pp. 57--93.
Abstract
For scattering systems consisting of a (family of) maximal dissipative extension(s) and a selfadjoint extension of a symmetric operator with finite deficiency indices, the spectral shift function is expressed in terms of an abstract Titchmarsh-Weyl function and a variant of the Birman-Krein formula is proved.

J. Behrndt, H. Neidhardt, J. Rehberg, Block matrices, optical potentials, trace class perturbations and scattering, in: Operator Theory in Inner Product Spaces, K.-H. Förster, P. Jonas, H. Langer, C. Trunk, eds., 175 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2007, pp. 33--49.

J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering systems and characteristic functions, in: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto, Japan, July 24--28, 2006, pp. 1940--1945.

J.F. Brasche, M.M. Malamud, H. Neidhardt, Selfadjoint extensions with several gaps: Finite deficiency indices, in: Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems, K.-H. Förster, P. Jonas, H. Langer, eds., 162 of Operator Theory, Advances and Applications, Birkhäuser, Basel, 2006, pp. 85-101.

T. Ichinose, H. Neidhardt, V.A. Zagrebnov, Operator norm convergence of Trotter-Kato product formula, in: Proceedings of the International Conference on Functional Analysis, Ukrainian Academic Press, Kiev, 2003, pp. 100--106.

H.-Chr. Kaiser, U. Bandelow, Th. Koprucki, J. Rehberg, Modelling and simulation of strained quantum wells in semiconductor lasers, in: Mathematics --- Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.-J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 377--390.

U. Bandelow, H. Gajewski, H.-Chr. Kaiser, Modeling combined effects of carrier injection, photon dynamics and heating in Strained Multi-Quantum-Well Laser, in: Physics and Simulation of Optoelectronic Devices VIII, R.H. Binder, P. Blood, M. Osinski, eds., 3944 of Proceedings of SPIE, SPIE, Bellingham, WA, 2000, pp. 301--310.

J.F. Brasche, M.M. Malamud, H. Neidhardt, Weyl functions and singular continuous spectra of self-adjoint extensions, in: Stochastic processes, physics and geometry: New interplays. II. A volume in honor of Sergio Albeverio. Proceedings of the conference on infinite dimensional (stochastic) analysis and quantum physics, Leipzig, Germany, January 18-22, 1999, F. Gesztesy, H. Holden, J. caps">caps">Jost et al., eds., 29 of CMS Conf. Proc., Amer. Math. Soc., Providence, 2000, pp. 75--84.

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

M. Hintermüller, S.-M. Stengl, On the convexity of optimal control problems involving non-linear PDEs or VIs and applications to Nash games, Preprint no. 2759, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2759 .
Abstract, PDF (381 kByte)
Generalized Nash equilibrium problems in function spaces involving PDEs are considered. One of the central issues arising in this context is the question of existence, which requires the topological characterization of the set of minimizers for each player of the associated Nash game. In this paper, we propose conditions on the operator and the functional that guarantee the reduced formulation to be a convex minimization problem. Subsequently, we generalize results of convex analysis to derive optimality systems also for non-smooth operators. Our theoretical findings are illustrated by examples.

R. Chill, H. Meinlschmidt, J. Rehberg, On the numerical range of second order elliptic operators with mixed boundary conditions in L^p, Preprint no. 2723, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2723 .
Abstract, PDF (247 kByte)
We consider second order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on L^p in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in [7]. Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin- instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterization of elements of the form domains inducing mixed boundary conditions.

M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, H. Sun, Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization, Preprint no. 2689, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2689 .
Abstract, PDF (25 MByte)
Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.

A.F.M. TER Elst, A. Linke, J. Rehberg, On the numerical range of sectorial forms, Preprint no. 2667, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2667 .
Abstract, PDF (245 kByte)
We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper H^∞-angle for the H^∞-calculus on L_p for all p ∈ (1, ∞) if the coefficients are real valued.

D. Bothe, P.-É. Druet, Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models, Preprint no. 2658, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2658 .
Abstract, PDF (509 kByte)
We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier- Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.

H. Stephan, Millions of Perrin pseudoprimes including a few giants, Preprint no. 2657, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2657 .
Abstract, PDF (244 kByte)
The calculation of many and large Perrin pseudoprimes is a challenge. This is mainly due to their rarity. Perrin pseudoprimes are one of the rarest known pseudoprimes. In order to calculate many such large numbers, one needs not only a fast algorithm but also an idea how most of them are structured to minimize the amount of numbers one have to test. We present a quick algorithm for testing Perrin pseudoprimes and develop some ideas on how Perrin pseudoprimes might be structured. This leads to some conjectures that still need to be proved.
We think that we have found well over 90% of all 20-digit Perrin pseudoprimes. Overall, we have been able to calculate over 9 million Perrin pseudoprimes with our method, including some very large ones. The largest number found has 1436 digits. This seems to be a breakthrough, compared to the previously known just over 100,000 Perrin pseudoprimes, of which the largest have 20 digits.
In addition, we propose two sequences that do not provide any pseudoprimes up to 1,000,000,000 at all.

D. Gahururu, M. Hintermüller, S.-M. Stengl, Th.M. Surowiec, Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion, Preprint no. 2654, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2654 .
Abstract, PDF (337 kByte)
PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs.

M. Hintermüller, K. Papafitsoros, C.N. Rautenberg, Variable step mollifiers and applications, Preprint no. 2628, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2628 .
Abstract, PDF (562 kByte)
We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections.

M. Hintermüller, T. Keil, Optimal control of geometric partial differential equations, Preprint no. 2612, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2612 .
Abstract, PDF (15 MByte)
Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of nondegenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint.

A.F.M. TER Elst, R. Haller-Dintelmann, J. Rehberg, P. Tolksdorf, On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients, Preprint no. 2590, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2590 .
Abstract, PDF (383 kByte)
Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on L^p(Ω). Additional properties like analyticity of the semigroup, H^∞-calculus and maximal regularity arealso discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of ^p's for small imaginary parts of the coefficients. Our results are based on the recent notion of ^p-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.

A.F.M. TER Elst, H. Meinlschmidt, J. Rehberg, Essential boundedness for solutions of the Neumann problem on general domains, Preprint no. 2574, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2574 .
Abstract, PDF (220 kByte)
Let the domain under consideration be bounded. Under the suppositions of very weak Sobolev embeddings we prove that the solutions of the Neumann problem for an elliptic, second order divergence operator are essentially bounded, if the right hand sides are taken from the dual of a Sobolev space which is adapted to the above embedding.

A. Mielke, J. Naumann, On the existence of global-in-time weak solutions and scaling laws for Kolmogorov's two-equation model of turbulence, Preprint no. 2545, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2545 .
Abstract, PDF (467 kByte)
This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.

G. Dong, Integrated physics-based method, learning-informed model and hyperbolic PDEs for imaging, Efficient Algorithms in Data Science, Learning and Computational Physics, Sanya, China, January 12 - 16, 2020.

A. Mielke, Variational structures for the analysis of PDE systems, Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (online), October 12 - 23, 2020, WIAS Berlin, October 13, 2020.

K. Papafitsoros, Automatic distributed regularization parameter selection in Total Generalized Variation image reconstruction via bilevel optimization, Seminar, Shenzhen MSU-BIT University, Department of Mathematics, Shenzhen, China, January 16, 2020.

K. Papafitsoros, Automatic distributed regularization parameter selection in imaging via bilevel optimization, Workshop on PDE Constrained Optimization under Uncertainty and Mean Field Games, January 28 - 30, 2020, WIAS, Berlin, January 30, 2020.

K. Papafitsoros, Spatially dependent parameter selection in TGV based image restoration via bilevel optimization, Efficient Algorithms in Data Science, Learning and Computational Physics, Sanya, China, January 12 - 16, 2020.

J. Rehberg, Explicit and uniform estimates for second order divergence operators on L^p spaces, Oberseminar ``Analysis und Theoretische Physik'', Leibniz Universität Hannover, Institut für Angewandte Mathematik, January 28, 2020.

A. Stephan, On evolution semigroups and Trotter product operator-norm estimates, Operator Theory and Krein Spaces, December 19 - 22, 2019, Technische Universität Wien, Austria, December 20, 2019.

W. van Zuijlen, Bochner integrals in ordered vector spaces, Analysis Seminar, University of Canterbury, Department of Mathematics and Statistics, UK, March 1, 2019.

W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, 12th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis, December 4 - 6, 2019, University of Oxford, Mathematical Institute, UK, December 6, 2019.

W. van Zuijlen, From periodic to Dirichlet and Neumann on boxes, Seminar Forschergruppe 2402: Research Unit --- Rough paths, stochastic partial differential equations and related topics, Technische Universität Berlin, Institut für Mathematik, December 12, 2019.

W. van Zuijlen, Mini-course on Besov spaces I--III, Junior Trimester Program: Randomness, PDEs and Nonlinear Fluctuations (Sept. 2 to Dec. 19, 2019), October 16 - November 6, 2019, Hausdorff Research Institute for Mathematics (HIM), Bonn.

M. Hintermüller, (Pre)Dualization, dense embeddings of convex sets, and applications in image processing, Thematic Programme ``Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions'', Workshop ``Numerical Algorithms in Nonsmooth Optimization'', February 25 - March 1, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, February 28, 2019.

M. Hintermüller, A function space framework for structural total variation regularization with applications in inverse problems, Thematic Programme ``Modern Maximal Monotone Operator Theory: From Nonsmooth Optimization to Differential Inclusions'', Workshop ``Nonsmooth and Variational Analysis'', January 28 - February 1, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, Austria, February 1, 2019.

M. Hintermüller, Structural total variation regularization with applications in inverse problems, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), DFG Priority Programme 1962 ``Non Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization'', February 18 - 22, 2019, Technische Universität Wien, Austria, February 19, 2019.

M. Liero, Effective diffusion in thin structures via generalized gradient systems and EDP-convergence, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18 - 22, 2019, Universität Wien, Technische Universität Wien, Austria, February 20, 2019.

A. Mielke, Transport versus growth and decay: The (spherical) Hellinger--Kantorovich distance between arbitrary measures, Optimal Transport: From Geometry to Numerics, May 13 - 17, 2019, Erwin Schrödinger International Institute for Mathematics and Physics, Universität Wien, Austria, May 17, 2019.

K. Papafitsoros, Quantitative MRI: From fingerprinting to integrated physics-based models, Synergistic Reconstruction Symposium, November 3 - 6, 2019, Chester, UK, November 4, 2019.

J. Rehberg, An extrapolation for the Lax--Milgram isomorphism for second order divergence operators, Oberseminar ``Angewandte Analysis'', Technische Universität Darmstadt, February 7, 2019.

J. Rehberg, Explicit and uniform estimates for second order divergence operators on $L^P$ spaces, Evolution Equations: Applied and Abstract Perspectives, October 28 - November 1, 2019, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, October 31, 2019.

J. Rehberg, Maximal parabolic regularity for the treatment of real world problems, 12th Workshop on Analysis and Advanced Numerical Methods for Partial Differential Equations (not only) for Junior Scientists (AANMPDE 12), July 1 - 5, 2019, Österreichische Akademie der Wissenschaften, St. Wolfgang / Strobl, Austria, July 2, 2019.

M. Heida, On convergences of the square root approximation scheme to the Fokker--Planck operator, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', February 11 - 17, 2018, Mathematisches Forschungsinstitut Oberwolfach, February 13, 2018.

S.-M. Stengl, Generalized Nash equilibrium problems with partial differential operators: theory, algorithms and risk aversion, Annual Meeting of the DFG Priority Programme 1962, October 1 - 3, 2018, Kremmen (Sommerfeld), October 1, 2018.

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

K. Papafitsoros, A function space framework for structural total variation regularization with applications in inverse problems, SIAM Conference on Imaging Science, Minisymposium MS38 ``Geometry-driven Anisotropic Approaches for Imaging Problems'', June 5 - 8, 2018, Bologna, Italy, June 6, 2018.

D.R.M. Renger, Gradient flows and GENERIC in flux space, Workshop ``Variational Methods for Evolution'', November 12 - 18, 2017, Mathematisches Forschungsinstitut Oberwolfach, November 16, 2017.

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

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

D.R.M. Renger, Large deviations and gradient flows, Spring School 2017: From Particle Dynamics to Gradient Flows, February 27 - March 3, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, March 1, 2017.

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

M. Hintermüller, (Pre)Dualization, dense embeddings of convex sets, and applications in image processing, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, International Center for Mathematics, University of Lisbon, Portugal, December 6, 2017.

M. Hintermüller, (Pre)Dualization, dense embeddings of convex sets, and applications in image processing, HCM Workshop: Nonsmooth Optimization and its Applications, May 15 - 19, 2017, Hausdorff Center for Mathematics, Bonn, May 15, 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 PDE constrained optimization, Optimization Seminar, Chinese Academy of Sciences, State Key Laboratory of Scientific and Engineering Computing, Beijing, China, June 6, 2017.

M. Hintermüller, On (pre)dualization, dense embeddings of convex sets, and applications in image processing, Seminar, Isaac Newton Institute, Programme ``Variational Methods and Effective Algorithms for Imaging and Vision'', Cambridge, UK, August 30, 2017.

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

M. Hintermüller, Optimal control of multiphase fluids and droplets, Kolloquium, Friedrich-Alexander-Universität Erlangen-Nürnberg, 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. Liero, The Hellinger--Kantorovich distance as natural generalization of optimal transport distance to (scalar) reaction-diffusion equations, Workshop ``Variational Methods for Evolution'', November 12 - 17, 2017, Mathematisches Forschungsinstitut Oberwolfach, November 14, 2017.

M. Liero, The Hellinger--Kantorovich distance as natural generalization of optimal transport distance to (scalar) reaction-diffusion equations, Oberseminar ``Angewandte Analysis'', Universität Dortmund, Institut für Mathematik, November 29, 2017.

M. Mittnenzweig, A variational approach to quantum master equations coupled to a semiconductor PDE, Workshop ``Variational Methods for Evolution'', November 12 - 17, 2017, Mathematisches Forschungsinstitut Oberwolfach, November 14, 2017.

J. Rehberg, Explicit and uniform resolvent estimates for second order divergence operators on $L^p$ spaces, Oberseminar Analysis, Technische Universität Darmstadt, Fachbereich Mathematik, November 9, 2017.

M. Heida, Homogenization of the random conductance model, 7th European Congress of Mathematics (ECM), session ``Probability, Statistics and Financial Mathematics'', July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

M. Heida, Homogenization of the random conductance model, Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 26 - 28, 2016, Technische Universität Dortmund, Fachbereich Mathematik, Dortmund, September 26, 2016.

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

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

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

M. Liero, On Entropy-Transport problems and distances between positive measures, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 25, 2016.

M. Liero, On entropy-transport problems and the Hellinger--Kantorovich distance, Follow-up Workshop to Junior Hausdorff Trimester Program ``Optimal Transportation'', August 29 - September 2, 2016, Hausdorff Research Institute for Mathematics, Bonn, August 30, 2016.

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

M. Mittnenzweig, Gradient structures for Lindblad equations satisfying detailed balance, 3rd PhD Workshop, May 30 - 31, 2016, International Research Training Group of the Collaborative Research Center (SFB) 1114 ``Scaling Cascades in Complex Systems'', Güstrow, May 31, 2016.

H. Neidhardt, To the spectral theory of vector-valued Sturm--Liouville operators with summable potentials and point interactions, Workshop ``Mathematical Challenge of Quantum Transport in Nanosystems'' (Pierre Duclos Workshop), November 14 - 15, 2016, Saint Petersburg National Research University of Informational Technologies, Mechanics, and Optics, Russian Federation, November 15, 2016.

D.R.M. Renger, Functions of bounded variation with an infinite-dimensional codomain, Meeting in Applied Mathematics and Calculus of Variations, September 13 - 16, 2016, Università di Roma ``La Sapienza'', Dipartimento di Matematica ``Guido Castelnuovo'', Italy, September 16, 2016.

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

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

M. Hintermüller, Nonsmooth structures in PDE constrained optimization, 66th Workshop ``Advances in Convex Analysis and Optimization'', July 5 - 10, 2016, International Centre for Scientific Culture ``E. Majorana'', School of Mathematics ``G. Stampacchia'', Erice, Italy, July 9, 2016.

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

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

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

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

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

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

D.R.M. Renger, Large deviations for reacting particle systems: The empirical and ensemble processes, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, July 30, 2015.

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

D.R.M. Renger, The empirical process of reacting particles: Large deviations and thermodynamic principles, Minisymposium ``Real World Phenomena Explained by Microscopic Particle Models'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 8 - 22, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 10, 2015.

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

H. Neidhardt, Boundary triplet approach and tensor products, Seminar ``Angewandte Analysis und Numerische Mathematik'', Technische Universität Graz, Institut für Numerische Mathematik, Graz, Austria, January 29, 2015.

H. Neidhardt, Boundary triplets and trace formulas, Workshop ``Spectral Theory and Weyl Function'', January 5 - 9, 2015, Mathematisches Forschungsinstitut Oberwolfach, January 5, 2015.

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

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

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

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

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

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

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

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

H. Neidhardt, Trace formula for extensions, Workshop on Linear Relations and Extension Theory, September 16 - 19, 2014, Technische Universität Graz, Institut für Numerische Mathematik, Obergurgl, Austria, September 18, 2014.

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

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

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

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

H. Neidhardt, Perturbation determinants for singular perturbations, Séminaire Analyse et Géometrie, Université d'Aix-Marseille, Centre de Mathématiques et Informatique, Marseille, France, May 13, 2013.

H. Neidhardt, Sturm--Liouville boundary value problems with operator potentials, System and Operator Realizations of Analytic Functions, February 18 - 22, 2013, Lorentz Center, Leiden, Netherlands, February 22, 2013.

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

H. Neidhardt, An application of the Landauer--Büttiker formula to photon emitting and absorbing systems, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24 - 28, 2012, WIAS Berlin, September 28, 2012.

H. Neidhardt, Jaynes--Cummings model coupled to leads: A model for LEDs?, Quantum Circle Seminar, Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Doppler Institute for Mathematical Physics and Applied Mathematics, Prague, Czech Republic, March 13, 2012.

H. Neidhardt, Landauer--Büttiker formula applied to photon emitting and absorbing systems, Kolloquium ``Mathematische Physik'', December 13 - 14, 2012, Technische Universität Clausthal/Technische Universität Braunschweig, Clausthal-Zellerfeld, December 14, 2012.

H. Neidhardt, On the abstract Landauer--Buettiker formula and applications, Workshop on Spectral Theory and Differential Operators, August 27 - 31, 2012, Technische Universität Graz, Institut für Numerische Mathematik, Austria, August 30, 2012.

TH. Koprucki, Semi-classical modeling of quantum dot lasers with microscopic treatment of Coulomb scattering, Mathematical Challenges of Quantum Transport in Nano-Optoelectronic Systems, February 4 - 5, 2011, WIAS, February 4, 2011.

L. Wilhelm, An abstract Landauer--Büttiker formula with application to a toy model of a quantum dot LED, Analysis Seminar, Aalborg University, Department of Mathematical Sciences, Denmark, June 16, 2011.

L. Wilhelm, An abstract approach to the Landauer--Büttiker formula with application to an LED toy model, Mathematical Challenges of Quantum Transport in Nano-Optoelectronic Systems, February 4 - 5, 2011, WIAS, February 5, 2011.

H. Neidhardt, Boundary triplets and a result of Jost and Pais: An abstract approach, Mathematical Results in Quantum Physics (QMath11), Topical Sesson ``Spectral Theory'', September 6 - 10, 2010, University of Hradec Králové, Czech Republic, September 8, 2010.

H. Neidhardt, Extensions, perturbation determinants and trace formulas, Workshop on Systems & Operators in Honor of Henk de Snoo, December 14 - 17, 2010, University of Groningen, Faculty of Mathematics and Natural Sciences, Netherlands, December 16, 2010.

H. Neidhardt, On perturbation determinants for non-selfadjoint operators, ESF Exploratory Workshop on Mathematical Aspects of the Physics with Non-Self-Adjoint Operators, August 30 - September 3, 2010, Czech Academy of Sciences, Nuclear Physics Institute, Prague, September 1, 2010.

H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, 21th International Workshop on Operator Theory and Applications (IWOTA 2010), Session ``Extension Theory and Applications'', July 12 - 16, 2010, Technische Universität Berlin, Institut für Mathematik, July 15, 2010.

H. Neidhardt, Scattering for self-adjoint extensions, Analysis Seminar, Aalborg University, Department of Mathematical Sciences, Denmark, April 22, 2010.

H. Neidhardt, Scattering matrices and Weyl function, Research seminar of the Graduate School of Natural Science and Technology, Okayama University, Department of Mathematics, Japan, September 14, 2010.

H. Neidhardt, The contribution of Takashi to Trotter--Kato product formulas for imaginary times, Symposium dedicated to Takashi Ichinose's 70th birthday, Kanazawa University, Faculty of Science, Department of Mathematics, Japan, September 17, 2010.

J. Rehberg, $L^infty$-estimates for fractional resolvent powers, 21th International Workshop on Operator Theory and Applications (IWOTA 2010), Session ``Spectral Theory and Evolution Equations'', July 12 - 16, 2010, Technische Universität Berlin, Institut für Mathematik, July 13, 2010.

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

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

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

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

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

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

H. Neidhardt, On the Kato--Rosenblum and the Weyl--von Neumann theorem for self-adjoint extensions of symmetric operators, Séminaire de Physique Statistique & Matière Condensée, Centre National de la Recherche Scientifique Luminy, Centre de Physique Théorique, Marseille, France, October 7, 2009.

H. Neidhardt, Sturm--Liouville operators with operator potentials, Quantum Circle Seminar, Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Prague, March 17, 2009.

H. Neidhardt, Trotter--Kato product formula for imaginary times, International Congress of Mathematical Physics, Prague, Czech Republic, August 3 - 8, 2009.

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

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

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

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

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

K. Hoke, Numerical treatment of the Kohn--Sham system for semiconductor devices, Workshop on Mathematical Aspects of Transport in Mesoscopic Systems, Dublin, Ireland, December 4 - 7, 2008.

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

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

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

H. Neidhardt, On the trace formula for pairs of extensions, Mathematical Physics and Spectral Theory, April 24 - 26, 2008, Humboldt-Universität zu Berlin, Workshop in Memory of Vladimir Geyler, April 26, 2008.

H. Neidhardt, On the unitary equivalence of absolutely continuous parts of self-adjoint extensions, 8th Workshop on Operator Theory in Krein Spaces and Inverse Problems, December 18 - 21, 2008, Technische Universität Berlin, Institut für Mathematik, December 19, 2008.

H. Neidhardt, On the unitary equivalence of the absolutely continuous parts of self-adjoint extensions, Annual Meeting of the Deutsche Mathematiker-Vereinigung 2008, Minisymposium ``Operatortheorie'', September 15 - 19, 2008, Friedrich-Alexander-Universität Erlangen-Nürnberg, September 19, 2008.

H. Neidhardt, Scattering theory for open quantum systems, Centre National de la Recherche Scientifique, Luminy, Centre de Physique Théorique, Marseille, France, October 22, 2008.

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

G. Schmidt, Approximation of pseudodifferential and integral operators, Conference ``Analysis, PDEs and Applications'', June 30 - July 3, 2008, Rome, Italy, July 3, 2008.

G. Schmidt, Approximation of pseudodifferential and integral operators, University of Bath, Department of Mathematical Sciences, UK, December 5, 2008.

K. Hoke, The Kohn--Sham system in case of zero temperature, Mini-Workshop on PDE's and Quantum Transport, March 12 - 16, 2007, Aalborg University, Department of Mathematical Sciences, Denmark, March 15, 2007.

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

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

H. Neidhardt, Boundary triplets and scattering, International Conference ``Modern Analysis and Applications'' (MAA 2007), April 9 - 14, 2007, Institute of Mathematics, Economics and Mechanics of Odessa National I.I. Mechnikov University, Ukraine, April 13, 2007.

H. Neidhardt, On Eisenbud's and Wigner's $R$-matrix: A general approach, 10th Quantum Mathematics International Conference (QMath 10), September 10 - 15, 2007, Institute of Mathematics ``Simion Stoilow'' of the Romanian Academy, Moeciu, Romania, September 11, 2007.

H. Neidhardt, On trace formula and Birman--Krein formula for pairs of extensions, 7th Workshop on Operator Theory in Krein Spaces and Spectral Analysis, December 13 - 16, 2007, Technische Universität Berlin, Institut für Mathematik, December 13, 2007.

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

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

J. Rehberg, On Schrödinger--Poisson systems, International Conference ``Nonlinear Partial Differential Equations'' (NPDE 2007), September 10 - 15, 2007, Institute of Applied Mathematics and Mechanics of NASU, Yalta, Ukraine, September 13, 2007.

J. Rehberg, Operator functions inherit monotonicity, Mini-Workshop on PDE's and Quantum Transport, March 12 - 16, 2007, Aalborg University, Department of Mathematical Sciences, Denmark, March 14, 2007.

J. Rehberg, Über Schrödinger-Poisson-Systeme, Chemnitzer Mathematisches Colloquium, Technische Universität Chemnitz, Fakultät für Mathematik, May 24, 2007.

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

G. Schmidt, Regularity of solutions to anisotropic elliptic transmission problems, Fifth Singular Days, April 23 - 27, 2007, International Center for Mathematical Meetings, Luminy, France, April 24, 2007.

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

H. Neidhardt, Lax--Phillips scattering revisited, Operator Theory, Analysis and Mathematical Physics (OTAMP 2006), June 15 - 22, 2006, Lund University, Lund, Sweden, June 20, 2006.

H. Neidhardt, Open quantum systems and scattering, Operator Theory in Quantum Physics, September 9 - 14, 2006, Nuclear Physics Institute of the Academy of Sciences, Prague, Czech Republic, September 10, 2006.

H. Neidhardt, Wigner's $R$-matrix and Weyl functions, 6th Workshop ``Operator Theory in Krein Spaces and Operator Polynomials'', December 14 - 17, 2006, Technische Universität Berlin, December 14, 2006.

J. Rehberg, Existence and uniqueness for van Roosbroeck's system in Lebesque spaces, Conference ``Recent Advances in Nonlinear Partial Differential Equations and Applications'', Toledo, Spain, June 7 - 10, 2006.

J. Rehberg, Regularity for nonsmooth elliptic problems, Crimean Autumn Mathematical School, September 20 - 25, 2006, Vernadskiy Tavricheskiy National University, Laspi, Ukraine, September 21, 2006.

J. Rehberg, The Schrödinger--Poisson system, Colloquium in Honor of Prof. Demuth, September 10 - 11, 2006, Universität Clausthal, September 10, 2006.

G. Schmidt, Regularity of solutions to anisotropic elliptic transmission problems, University of Liverpool, Department of Mathematical Sciences, UK, October 25, 2006.

H. Neidhardt, Block matrices, boundary triplets and scattering, 5th Workshop ``Operator Theory in Krein Spaces and Differential Equations'', December 16 - 18, 2005, Technische Universität Berlin, December 17, 2005.

H. Neidhardt, Zeno product formula and continual observations in quantum mechanics, Workshop on Operator Semigroups, Evolution Equations, and Spectral Theory in Mathematical Physics, October 3 - 7, 2005, Centre International de Rencontres Mathématiques, Marseille, France, October 5, 2005.

H.-Chr. Kaiser, About quantum transmission on an up to three dimensional spatial domain, University of Texas at Dallas, USA, October 28, 2005.

H.-Chr. Kaiser, An open quantum system driven by an external flow, Workshop ``Nonlinear spectral problems in solid state physics'', April 4 - 8, 2005, Institut Henri Poincaré, Paris, France, April 7, 2005.

H.-Chr. Kaiser, Modeling and quasi-3D simulation of indium grains in (In,Ga)N/GaN quantum wells by means of density functional theory, Physikalisches Kolloquium, Brandenburgische Technische Universität, Lehrstuhl Theoretische Physik, Cottbus, February 15, 2005.

H.-Chr. Kaiser, On quantum transmission, Mathematical Physics Seminar, November 9 - 11, 2005, University of Texas at Austin, USA, November 9, 2005.

H.-Chr. Kaiser, Quasi-3D simulation of multi-excitons by means of density functional theory, Oberseminar ``Numerik/Wissenschaftliches Rechnen'', Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 11, 2005.

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

J. Rehberg, Elliptische und parabolische Probleme aus Anwendungen, Kolloquium im Fachbereich Mathematik, Universität Darmstadt, May 18, 2005.

J. Rehberg, Existence, uniqeness and regularity for quasilinear parabolic systems, International Conference ``Nonlinear Partial Differential Equations'', September 17 - 24, 2005, Institute of Applied Mathematics and Mechanics Donetsk, Alushta, Ukraine, September 18, 2005.

J. Rehberg, H$^1,q$-regularity for linear, elliptic boundary value problems, Regularity for nonlinear and linear PDEs in nonsmooth domains - Analysis, simulation and application, September 5 - 7, 2005, Universität Stuttgart, Deutsche Forschungsgemeinschaft (SFB 404), Hirschegg, Austria, September 6, 2005.

J. Rehberg, Regularität für elliptische Probleme mit unglatten Daten, Oberseminar Prof. Escher/Prof. Schrohe, Technische Universität Hannover, December 13, 2005.

J. Rehberg, Some analytical ideas concerning the quantum-drift-diffusion systems, Workshop ``Problèmes spectraux non-linéaires et modèles de champs moyens'', April 4 - 8, 2005, Institut Henri Poincaré, Paris, France, April 5, 2005.

J. Rehberg, Analysis of macroscopic and quantum mechanical semiconductor models, International Visitor Program ``Nonlinear Parabolic Problems'', August 8 - November 18, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, November 1, 2005.

J. Rehberg, Existence, uniqueness and regularity for quasilinear parabolic systems, Conference ``Nonlinear Parabolic Problems'', October 17 - 21, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, October 20, 2005.

G. Schmidt, Corner singularities for anisotropic transmission problems, International Conference ``Nonlinear Partial Differential Equations'' (NPDE2005), September 17 - 23, 2005, Alushta, Ukraine, September 20, 2005.

H. Neidhardt, Hybrid models for semiconductors, QMath9, September 12 - 16, 2004, Giens, France, September 12, 2004.

H. Neidhardt, Zeno product formula, 4th Workshop ``Operator Theory in Krein Spaces and Applications'', December 17 - 19, 2004, Technische Universität Berlin, December 19, 2004.

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

J. Rehberg, Elliptische und parabolische Probleme mit unglatten Daten, Technische Universität Darmstadt, Fachbereich Mathematik, December 14, 2004.

J. Rehberg, Quasilinear parabolic equations in $L^p$, Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann, June 28 - 30, 2004, Universität Zürich, Institut für Mathematik, Switzerland, June 29, 2004.

J. Rehberg, The two-dimensional van Roosbroeck system has solutions in $L^p$, Workshop ``Advances in Mathematical Semiconductor Modelling: Devices and Circuits'', March 2 - 6, 2004, Chinese-German Centre for Science Promotion, Beijing, China, March 5, 2004.

H. Neidhardt, Self-adjoint extensions with several gaps: Scalar-type Weyl functions, 3rd Workshop Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems, December 12 - 14, 2003, Technische Universität Berlin, Institut für Mathematik, December 14, 2003.

J. Rehberg, A combined quantum mechanical and macroscopic model for semiconductors, Workshop on Multiscale problems in quantum mechanics and averaging techniques, December 11 - 12, 2003, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, December 12, 2003.

J. Rehberg, Solvability and regularity for parabolic equations with nonsmooth data, International Conference ``Nonlinear Partial Differential Equations'', September 15 - 21, 2003, National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics, Alushta, September 17, 2003.

J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering matrices and Dirichlet-to-Neumann maps, Preprint no. arXiv:1511.02376v2, Cornell University Library, arXiv.org, 2016.
Abstract
A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh-Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.

J. Behrndt, M.M. Malamud, H. Neidhardt, Scattering matrices and Dirichlet-to-Neumann maps, Preprint no. arXiv:1510.06219, Cornell University Library, arXiv.org, 2015.
Abstract
A general representation formula for the scattering matrix of a scattering system consisting of two selfadjoint operators in terms of an abstract operator valued Titchmarsh-Weyl m-function is proved. This result is applicable to scattering problems for different selfadjoint realizations of Schrödinger operators on unbounded domains, and Schrödinger operators with singular potentials supported on hypersurfaces. In both applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.

S. Albeverio, A. Kostenko, M. Malamud, H. Neidhardt, Schrödinger operators with concentric $delta$-shells, Preprint no. arXiv:1211.4048, Cornell University Library, arXiv.org, 2012.

H. Cornean, H. Neidhardt, L. Wilhelm, V. Zagrebnov, Cayley transform applied to non-interacting quantum transport, Preprint no. arXiv:1212.4965, Cornell University Library, arXiv.org, 2012.

M.M. Malamud, H. Neidhardt, Perturbation determinants and trace formulas for singular perturbations, Preprint no. arXiv:1212.6887, Cornell University Library, arXiv.org, 2012.

M. Baro, M. Demuth, E. Giere, Stable continuous spectra for differential operators of arbitrary order, Preprint no. 6, Technische Universität Clausthal, Institut für Mathematik, 2002.