On isoperimetric inequalities with respect to infinite measures

Abstract

We study isoperimetric problems with respect to infinite measures on $\mathbb{R} ^n$. 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 Pólya–Szegö-type inequalities, Sobolev embedding theorems, and a comparison result for elliptic boundary value problems.