Calculus of Variations and Geometric Measure Theory

E. Cinti - A. Pratelli

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

created by pratelli on 12 Sep 2012
modified on 15 Mar 2021


Published Paper

Inserted: 12 sep 2012
Last Updated: 15 mar 2021

Journal: J. Reine Angew. Math. (Crelle)
Year: 2017


We show that every isoperimetric set in $\mathbb R^N$ 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 ``$\varepsilon-\varepsilon$'' property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an ``$\varepsilon-\varepsilon^\beta$'' 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.