Calculus of Variations and Geometric Measure Theory

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 on 21 Mar 2018


Published Paper

Inserted: 19 nov 2014
Last Updated: 21 mar 2018

Journal: Journal of Geometric Analysis
Year: 2016

ArXiv: 1411.5208 PDF


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 $[$10$]$.