Weighted a Priori Estimates for the Solution of the Dirichlet Problem in Polygonal Domains in \(\mathbb{R}^2\)

Abstract

Let \(\Omega\) be a polygonal domain in \(\mathbb{R}^2\) and let \(U\) be a weak solution of \( -\Delta u=f\) in \( \Omega\) with Dirichlet boundary condition, where \(f\in L^p_\omega(\Omega)\) and \(\omega\) is a weight in \(A_p(\mathbb{R}^2)\), \(1<p<\infty\). We give some estimates of the Green function associated to this problem involving some functions of the distance to the vertices and the angles of \(\Omega\). As a consequence, we can prove an a priori estimate for the solution \(u\) on the weighted Sobolev spaces \(W^{2,p}_\omega(\Omega)\), \(1<p<\infty\).