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

TL;DR

This paper proves the existence of isoperimetric sets in Euclidean space with densities that are lower semi-continuous, converge from below to a positive limit at infinity, and are smaller than this limit far from the origin, confirming a conjecture for radial increasing densities.

Contribution

It establishes the existence of isoperimetric sets under specific density conditions, extending previous results and confirming a conjecture for radial increasing densities.

Findings

Existence of isoperimetric sets under specified density conditions

Sharpness of the main result demonstrated through known examples

Positive answer to a conjecture for radial increasing densities

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].