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- ItemHölder-estimates for non-autonomous parabolic problems with rough data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Meinlschmidt, Hannes; Rehberg, JoachimIn this paper we establish Hölder estimates for solutions to non-autonomous parabolic equations on non-smooth domains which are complemented with mixed boundary conditions. The corresponding elliptic operators are of divergence type, the coefficient matrix of which depends only measurably on time. These results are in the tradition of the classical book of Ladyshenskaya et al., which also serves as the starting point for our investigations.
- ItemOn the numerical range of second order elliptic operators with mixed boundary conditions in L$^p$(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Chill, Ralph; Meinlschmidt, Hannes; Rehberg, JoachimWe 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 Lp 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.
- ItemExtrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Meinlschmidt, Hannes; Rehberg, JoachimIn this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.
- ItemThe full Keller-Segel model is well-posed on fairly general domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Horstmann, Dirk; Rehberg, Joachim; Meinlschmidt, HannesIn 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. Nous considèrons le système de Keller et Segel dans son intégralité, un système quasilinéaire à réaction-diffusion fortement couplé. Le résultat principal montre que ce syst`eme est bien posé, cest-à-dire il admet une solution unique existant localement en temps à valeurs dans un espace fonctionnel approprié, pourvu que les valeurs initiales sont réguliers. Apparemment, il nexiste pas encore des résultats comparables. Pour la demonstration, nous utilisons des résultats récents de régularité elliptique et parabolique applicable à des domaines assez générals, combiné avec un théorème abstrait dAmann concernant les équations quasilinéaires non locales.
- ItemEssential boundedness for solutions of the Neumann problem on general domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) ter Elst, A.F.M.; Meinlschmidt, Hannes; Rehberg, JoachimLet 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.
- ItemRegularization for optimal control problems associated to nonlinear evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Meinlschmidt, Hannes; Meyer, Christian; Rehberg, JoachimIt is well-known that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard ``calculus of variations'' proof for the existence of optimal controls. For time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type spaces. In this paper, we propose an abstract function space and a suitable regularization- or Tychonov term for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacle-type in time. We establish the necessary properties of the function space and the Tychonov term and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand.
- ItemOptimal control of the thermistor problem in three spatial dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Meinlschmidt, Hannes; Meyer, Christian; Rehberg, JoachimThis 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 Pr¨uss. 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