Inserted: 19 nov 2014
Last Updated: 26 feb 2017
Journal: Journal of Geometric Analysis
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 $[$10$]$.