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search.filters.applied.f.access: openAccess × Author: Rehberg, Joachim × End date: 2022 × Start date: 2020 × Start date: 2000 × DDC: 510 × Has files: Yes × Type: Text × Version: publishedVersion × WGL Institute: WIAS ×

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Now showing 1 - 6 of 6
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    On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) ter Elst, A.F.M.; Haller-Dintelmann, Robert; Rehberg, Joachim; Tolksdorf, Patrick
    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 Lp(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.
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    Essential 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, Joachim
    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.
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    Regularization for optimal control problems associated to nonlinear evolution equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Meinlschmidt, Hannes; Meyer, Christian; Rehberg, Joachim
    It 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.
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    On the numerical range of sectorial forms
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) ter Elst, A.F.M.; Linke, Alexander; Rehberg, Joachim
    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 Lp for all p ∈ (1, ∞) if the coefficients are real valued.
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    On 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, Joachim
    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 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.
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    Extrapolated elliptic regularity and application to the van Roosbroeck system of semiconductor equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Meinlschmidt, Hannes; Rehberg, Joachim
    In 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.