# Revista Matemática Iberoamericana

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**Volume 29, Issue 2, 2013, pp. 665–690**

**DOI: 10.4171/RMI/734**

Published online: 2013-04-22

On isoperimetric inequalities with respect to infinite measures

Friedemann Brock^{[1]}, Anna Mercaldo

^{[2]}and Maria Rosaria Posteraro

^{[3]}(1) Universität Leipzig, Germany

(2) Università degli Studi di Napoli “Federico II”, Italy

(3) Università degli Studi di Napoli “Federico II”, Italy

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.

*Keywords: *Isoperimetric inequalities, infinite measures, Steiner symmetrization, Schwarz symmetrization, comparison result, Pólya–Szegö inequality

Brock Friedemann, Mercaldo Anna, Posteraro Maria Rosaria: On isoperimetric inequalities with respect to infinite measures . *Rev. Mat. Iberoamericana* 29 (2013), 665-690. doi: 10.4171/RMI/734