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 $μ$ defined by $dμ= e^{c|x|^2} dx$, $c\geq 0$, we prove that, among all sets with given $μ-$measure, the ball centered at the origin has the smallest (weighted) $μ-$perimeter. Our results are then applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems and a comparison result for elliptic boundary value problems.

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