Weighted estimates in $L^{2}$ for Laplace’s equation on Lipschitz domains

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 $\omega _{\alpha }(Q) =|Q-Q_{0}|^{\alpha }$, $Q_{0}$ 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.