article

Title: Weighted a priori estimates for the Poisson equation

Authors: Richard G. Duran, Marcela Sanmartino and Marisa Toschi

Issue: Volume 57 (2008), Issue 7, 3463-3478

Abstract:

$Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $\partial\Omega \in C^2$ and let $u$ be a solution of the classical Poisson problem in $\Omega$; i.e., \[ \begin{cases} -\Delta u = f &\mbox{in }\Omega,\\ u = 0 &\mbox{on }\partial\Omega, \end{cases} \] where $f \in L^p_{\omega}(\Omega)$ and $\omega$ is a weight in $A_p$. The main goal of this paper is to prove the following a priori estimate \[ \|u\|_{W^{2,p}_{\omega}(\Omega)} \le C\|f\|_{L^p_{\omega}(\Omega)}, \] and to give some applications for weights given by powers of the distance to the boundary.$