Weighted estimates in for Laplace's equation on Lipschitz domains

Author:
Zhongwei Shen

Journal:
Trans. Amer. Math. Soc. **357** (2005), 2843-2870

MSC (2000):
Primary 35J25

DOI:
https://doi.org/10.1090/S0002-9947-04-03608-6

Published electronically:
October 28, 2004

MathSciNet review:
2139930

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Abstract | References | Similar Articles | Additional Information

Abstract: Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

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Additional Information

**Zhongwei Shen**

Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Email:
shenz@ms.uky.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03608-6

Keywords:
Laplace equation,
Lipschitz domains,
weighted estimates

Received by editor(s):
October 20, 2002

Received by editor(s) in revised form:
December 11, 2003

Published electronically:
October 28, 2004

Article copyright:
© Copyright 2004
American Mathematical Society