Weighted a priori estimates for Poisson eqaution

DURÁN, RICARDO; SANMARTINO, MARCELA; TOSCHI, MARISA

El Escorial, Madrid

Conferencia; 8th International Conference on Harmonic Analysis and Partial Differential Equation; 2008

Universidad Autónoma de Madrid

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{eqnarray*} \left\{\begin{array}{cc} -\Delta u=f&\mbox{ in }\Omega\\ u=0&\mbox{ on }\partial\Omega \end{array}\right. \end{eqnarray*} where $f\in L^p_\omega(\Omega)$ and $\omega$ is a weight in $A_p$. The main goal of this work is to prove the following a priori estimate \begin{eqnarray*} \|u\|_{W^{2,p}_\omega(\Omega)} \le C\, \|f\|_{L^p_\omega(\Omega)}, \end{eqnarray*} and to give some applications for weights given by powers of the distance to the boundary.