Boundary value problems for the Laplacian in convex and semiconvex domains

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• 作者：Dorina Mitrea ; Marius Mitrea ; Lixin Yan
• 关键词：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
• 刊名：Journal of Functional Analysis
• 出版年：2010
• 出版时间：15 April 2010
• 年：2010
• 卷：258
• 期：8
• 页码：2507-2585
• 全文大小：630 K

文摘

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