Calculus of Variations and Geometric Measure Theory
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G. De Philippis - G. Franzina - A. Pratelli

Existence of isoperimetric sets with densities ''converging from below'' on $\mathbb R^N$

created by dephilipp on 19 Nov 2014
modified by pratelli on 30 Nov 2017


Published Paper

Inserted: 19 nov 2014
Last Updated: 30 nov 2017

Journal: Journal of Geometric Analysis
Year: 2016


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 limit $a>0$ at infinity, being $f\leq a$ 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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