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Weighted estimates in for Laplace's equation on Lipschitz domains

Authors:	Zhongwei Shen

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

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.

Keywords:	Laplace equation Lipschitz domains weighted estimates

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