On the isoperimetric problem with double density

TL;DR

This paper investigates the isoperimetric problem in Euclidean space with two different densities for volume and perimeter, proving the existence of optimal sets, thus generalizing previous single-density results.

Contribution

It establishes the existence of isoperimetric sets with double density, extending classical results to a more general setting with potential applications.

Findings

Existence of isoperimetric sets with double density proven.

Generalization of single-density isoperimetric results.

Framework applicable to various density configurations.

Abstract

In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to two different densities. The case of a single density, or equivalently, when the two densities coincide, has been well studied in the last years; nonetheless, the problem with two different densities is an important generalisation, also in view of applications. We will prove the existence of isoperimetric sets in this context, extending the known results for the case of single density.