Title: Weighted estimates for elliptic systems on Lipschitz domains

Authors: Zhongwei Shen

Issue: Volume 55 (2006), Issue 3, 1135-1154

Abstract: Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n \ge 3$. Let $\omega_{\lambda}(Q) = |Q-Q_0|^{\lambda}$, where $Q_0$ is a fixed point on $\partial\Omega$. For a second order elliptic system with constant coefficients on $\Omega$, we study boundary value problems with boundary data in the weighted space $L^2(\partial\Omega, \omega_{\lambda} d\sigma)$, where $d\sigma$ denotes the surface measure on $\partial\Omega$. We show that there exists $\varepsilon$ greater than $0$ such that the Dirichlet problem is uniquely solvable for $-\min(2{+}\varepsilon, n{-}1)$ less than $\lambda$ less than $\varepsilon$, and the Neumann type problem is uniquely solvable if $-\varepsilon$ less than $\lambda$ less than $\min(2{+}\varepsilon, n{-}1)$. The regularity for the Dirichlet problem with data in the weighted Sobolev space $L^2_1(\partial\Omega, \omega_{\lambda} d\sigma)$ is also considered.