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Elliptic and parabolic problems in non-smooth domains. (English) Zbl 1117.35015
Berlin: Logos Verlag; Darmstadt: TU Darmstadt, Fachbereich Mathematik (Dissertation) (ISBN 3-8325-1059-1/pbk). xii, 124 p. (2005).
The aim of this thesis is to investigate the regularity of solutions to PDEs in Lipschitz domains. These domains can have boundary singularities (corners or edges) which can lead to the solutions being less regular that one might expect from the data.
The known results for the Laplace’s equation for Lipschitz bounded domains are extended to unbounded Lipschitz domains.
The main result of this thesis is the maximum regularity for the Dirichlet-Laplacian and Neumann-Laplacian for the heat equation written in a Lipschitz domain.
Then, some operators with \(L^{\infty}\)-coefficients and Ornstein-Uhlenbeck operators are studied.
35B65 Smoothness and regularity of solutions to PDEs
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations