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On isoperimetric inequalities with respect to infinite measures

Abstract

We study isoperimetric problems with respect to infinite measures on RnR ^n. In the case of the measure μ\mu defined by dμ=ecx2dxd\mu = e^{c|x|^2} dx, c0c\geq 0, we prove that, among all sets with given μ\mu-measure, the ball centered at the origin has the smallest (weighted) μ\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.Comment: 25 page

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