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 y^k dxdy