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29 March 2024
 
  » arxiv » 1111.1468

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The mixed problem in Lipschitz domains with general decompositions of the boundary
Justin L. Taylor ; Katharine A. Ott ; Russell M. Brown ;
Date 7 Nov 2011
AbstractThis paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $Omegasubset eals^n$, $ngeq2$, with boundary that is decomposed as $partialOmega=Dcup N$, $D$ and $N$ disjoint. We let $Lambda$ denote the boundary of $D$ (relative to $partialOmega$) 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 $L^p(partialOmega)$. 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.
Source arXiv, 1111.1468
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