Boundary value problems for the Laplacian in convex and semiconvex domains

By D. Mitrea, M. Mitrea, Lixin Yan. 2010

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Abstract We study the fully inhomogeneous Dirichlet problem for the Laplacian in bounded convex domains in R n , 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).

Subfields: Applied Mathematics, Computational Theory and Mathematics

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