Weighted a priori estimates for the solution of the Dirichlet problem in polygonal domains in R2

Abstract

Let be a polygonal domain in R2 and let U be a weak solution of u = f in with Dirichlet boundary condition, where f 2 Lp !( ) and ! is a weight in Ap(R2), 1 < p < 1. 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 . As a consequence, we can prove an a priori estimate for the solution u on the weighted Sobolev spaces W2;p ! ( ), 1 < p < 1. 1 I