Boundary value problems for the Laplacian in convex and semiconvex domains |
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Authors: | Dorina Mitrea Marius Mitrea Lixin Yan |
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Affiliation: | a Department of Mathematics, University of Missouri, Columbia, MO 65211, USA b Department of Mathematics, Zhongshan University, Guangzhou, 510275, PR China |
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Abstract: | We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in Rn, when the size/smoothness of both the data and the solution are measured on scales of Besov and Triebel-Lizorkin spaces. As a preamble, we deal with the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates. As a corollary, sharp estimates for the Green potential are obtained in a variety of contexts, including local Hardy spaces. A substantial part of this analysis applies to bounded semiconvex domains (i.e., Lipschitz domains satisfying a uniform exterior ball condition). |
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Keywords: | Laplacian Semiconvex domain Convex domain Lipschitz domain satisfying a uniform exterior ball condition Besov and Triebel-Lizorkin spaces Nontangential maximal function Green operator Poisson problem |
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