The $\varepsilon-\varepsilon^\beta$ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets

TL;DR

This paper extends the $ ext{ extsterling}- ext{ extsterling}^eta$ property to the isoperimetric problem with double density, leading to new regularity results for isoperimetric sets, generalizing previous single density cases.

Contribution

It introduces the $ ext{ extsterling}- ext{ extsterling}^eta$ property for double density, a significant generalization from the single density case, and establishes regularity of isoperimetric sets under these conditions.

Findings

Proves the $ ext{ extsterling}- ext{ extsterling}^eta$ property for double density

Derives regularity results for isoperimetric sets with double density

Generalizes known properties from single to double density cases

Abstract

We prove the validity of the $\varepsilon-\varepsilon^\beta$ property in the isoperimetric problem with double density, generalising the known properties for the case of single density. As a consequence, we derive regularity for isoperimetric sets.