An isoperimetric inequality for Gauss--like product measures

Abstract

This paper deals with various questions related to the isoperimetic problem for smooth positive measure $d\mu =\phi (x)dx$, with $x\in \Omega \subset {\mathbb{R}}^N$. Firstly we find some necessary conditions on the density of the measure $\phi (x)$ that render the intersection of half spaces with $\Omega$ a minimum in the isoperimetric problem. We then identify the unique isoperimetric set for a wide class of factorized finite measures. These results are finally used in order to get sharp inequalities in weighted Sobolev spaces and a comparison result for solutions to boundary value problems for degenerate elliptic equations.