The mixed problem in Lipschitz domains with general decompositions of the boundary
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- by J. L. Taylor, K. A. Ott and R. M. Brown PDF
- Trans. Amer. Math. Soc. 365 (2013), 2895-2930 Request permission
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.References
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Additional Information
- J. L. Taylor
- Affiliation: Department of Mathematics, Murray State University, Murray, Kentucky 42071-3341
- Email: jtaylor52@murraystate.edu
- K. A. Ott
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
- MR Author ID: 810101
- Email: katharine.ott@uky.edu
- R. M. Brown
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
- MR Author ID: 259097
- Email: russell.brown@uky.edu
- 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. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 2895-2930
- MSC (2010): Primary 35J25, 35J05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05711-4
- MathSciNet review: 3034453