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Existence of isoperimetric sets with densities "converging from below" in $\mathbb{R}^N$

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arxiv 1411.5208 v1 pith:BLGJZOZT submitted 2014-11-19 math.AP

Guido De Philippis , Giovanni Franzina , Aldo Pratelli This is my paper
classification math.AP
keywords isoperimetricresultdensitiesdensitylimitmathbbsetsanswers
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abstract

In this paper, we consider the isoperimetric problem in the space $\mathbb{R}^N$ with density. Our result states that, if the density f is l.s.c. and converges to a positive limit at infinity, being smaller than this limit far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture made in [10].

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