The $\epsilon-\epsilon^\beta$ property, the boundedness of isoperimetric sets in $\R^N$ with density, and some applications

TL;DR

This paper proves that isoperimetric sets in Euclidean space with continuous, bounded density are necessarily bounded, using a weaker -^eta property, which relaxes previous Lipschitz requirements and has significant implications for their existence and regularity.

Contribution

It introduces a weaker -^eta property that ensures boundedness of isoperimetric sets under mere continuity of density, improving prior Lipschitz-based results.

Findings

Isoperimetric sets are bounded under continuous, bounded density.

A weaker -^eta property suffices for boundedness.

Results have implications for existence and regularity of isoperimetric sets.

Abstract

We show that every isoperimetric set in R^N with density is bounded if the density is continuous and bounded by above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical "\epsilon-\epsilon" property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an "\epsilon-\epsilon^\beta" version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the sole continuity is enough. We conclude by deriving some consequences of our result about the existence and regularity of isoperimetric sets.