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Transactions of the American Mathematical Society

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Weighted estimates in $L^{2}$ 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: Let $\Omega \subset \mathbb{R}^{d}$, $d\ge 3$, be a bounded Lipschitz domain. For Laplace's equation $\Delta u=0$ in $\Omega $, we study the Dirichlet and Neumann problems with boundary data in the weighted space $L^{2}(\partial \Omega ,\omega _{\alpha }d\sigma )$, where $\omega _{\alpha }(Q) =\vert Q-Q_{0}\vert^{\alpha }$, $Q_{0}$ is a fixed point on $\partial \Omega $, and $d\sigma $ denotes the surface measure on $\partial \Omega $. We prove that there exists $\varepsilon =\varepsilon (\Omega )\in (0,2]$ such that the Dirichlet problem is uniquely solvable if $1-d<\alpha <d-3+\varepsilon $, and the Neumann problem is uniquely solvable if $3-d-\varepsilon <\alpha <d-1$. If $\Omega $ is a $C^{1}$ domain, one may take $\varepsilon =2$. The regularity for the Dirichlet problem with data in the weighted Sobolev space $L^{2}_{1}(\partial \Omega ,\omega _{\alpha }d\sigma )$ is also considered. Finally we establish the weighted $L^{2}$ estimates with general $A_{p}$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

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