On isoperimetric inequalities with respect to infinite measures

We study isoperimetric problems with respect to infinite measures on $R^n$. In the case of the measure $\mu$ defined by $d\mu =e^{c\mathrm{\mid }x{\mathrm{\mid }}^2}dx$, $c\ge 0$, 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.