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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


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

Authors: J. L. Taylor, K. A. Ott and R. M. Brown
Journal: Trans. Amer. Math. Soc. 365 (2013), 2895-2930
MSC (2010): Primary 35J25, 35J05
Published electronically: December 13, 2012
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Abstract: This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $ \Omega \subset \mathbf {R}^n$, $ n\geq 2$, with boundary that is decomposed as $ \partial \Omega =D\cup N$, with $ 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 come from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.

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Additional Information

J. L. Taylor
Affiliation: Department of Mathematics, Murray State University, Murray, Kentucky 42071-3341

K. A. Ott
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

R. M. Brown
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

PII: S 0002-9947(2012)05711-4
Received by editor(s): May 10, 2011
Published electronically: December 13, 2012
Additional Notes: The second author’s research was supported in part by the National Science Foundation.
The third author’s research was supported in part by a grant from the Simons Foundation.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.