On weighted isoperimetric inequalities with non-radial densities

We consider a class of isoperimetric problems on ${\mathbb{R}}_{+}^N$ where the volume and the area element carry two different weights of the type $\mathrm{\mid }x{\mathrm{\mid }}^l{x}_N^{\alpha }$. We solve them in a special case while a more detailed study is contained in \cite{ABCMP2}. Our results imply a weighted Polya-Sz"ego principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations