Sharp estimates for solutions to a certain type of singular elliptic boundary value problems in two dimensions

Abstract

We consider the problem where G⊂R2 + is a bounded domain, adjacent to the x-axis, k is a real positive number, A is a positive definite 2x2 matrix with L∞ entries. Using symmetrization techniques, we get the sharp form of some estimates for u. The main tool is an isoperimetric inequality in the upper half plane with respect to the measure yk dxdy