On isoperimetric inequalities with respect to infinite measures

TL;DR

This paper investigates isoperimetric inequalities under infinite measures, proving that balls centered at the origin minimize perimeter for a class of exponential measures, with applications to inequalities and PDEs.

Contribution

It establishes isoperimetric inequalities for measures with exponential weights, extending classical results to infinite measure settings and deriving related functional inequalities.

Findings

Balls centered at the origin minimize perimeter for the measure e^{c|x|^2}

Derived Polya-Szego inequalities and Sobolev embeddings

Provided comparison results for elliptic boundary value problems

Abstract

We study isoperimetric problems with respect to infinite measures on $R ^n$. In the case of the measure $\mu$ defined by $d\mu = e^{c|x|^2} dx$, $c\geq 0$, we prove that, among all sets with given $\mu-$measure, the ball centered at the origin has the smallest (weighted) $\mu-$perimeter. Our results are then applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems and a comparison result for elliptic boundary value problems.