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Article overview
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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 | Abstract: | This 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 | Services: | Forum | Review | PDF | Favorites |
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