An isoperimetric inequality for Gauss--like product measures

TL;DR

This paper investigates isoperimetric inequalities for smooth measures in Euclidean spaces, identifying conditions for minimal boundary measures, characterizing unique minimizers for certain measures, and applying results to weighted Sobolev inequalities and elliptic PDEs.

Contribution

It establishes necessary conditions for measure densities to minimize boundary measures, characterizes unique isoperimetric sets for factorized measures, and applies findings to inequalities and PDE comparison results.

Findings

Half-space intersections minimize boundary measure under certain conditions

Unique isoperimetric sets identified for a class of factorized measures

Sharp weighted Sobolev inequalities derived and PDE comparison results obtained

Abstract

This paper deals with various questions related to the isoperimetic problem for smooth positive measure $d\mu = \varphi(x)dx$, with $x \in \Omega \subset \mathbb{R}^N$. Firstly we find some necessary conditions on the density of the measure $ \varphi(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.