The mixed problem in Lipschitz domains with general decompositions of the boundary

• Published in: arXiv.org 2011-11
• Main Authors: Taylor, Justin L, Ott, Katharine A, Brown, Russell M
• Format: Article
• Language: English
• Subjects: Decomposition; Dirichlet problem; Domains; Sobolev space
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1111.1468

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain \(\Omega\subset \reals^n\), \(n\geq2\), with boundary that is decomposed as \(\partial\Omega=D\cup N\), \(D\) and \(N\) disjoint. We let \(\Lambda\) denote the boundary of \(D\) (relative to \(\partial\Omega\)) and impose conditions on the dimension and shape of \(\Lambda\) and the sets \(N\) and \(D\). Under these geometric criteria, we show that there exists \(p_0>1\) depending on the domain \(\Omega\) such that for \(p\) in the interval \((1,p_0)\), the mixed problem with Neumann data in the space \(L^p(N)\) and Dirichlet data in the Sobolev space \(W^ {1,p}(D) \) has a unique solution with the non-tangential maximal function of the gradient of the solution in \(L^p(\partial\Omega)\). We also obtain results for \(p=1\) when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.