Calculus of Variations and Geometric Measure Theory

G. Saracco

A sufficient criterion to determine planar self-Cheeger sets

created by saracco on 28 Jun 2019
modified on 21 Sep 2021


Published Paper

Inserted: 28 jun 2019
Last Updated: 21 sep 2021

Journal: J. Convex Anal.
Volume: 28
Number: 3
Pages: 951--958
Year: 2021

ArXiv: 1906.12101 PDF
Links: Published version on J. Convex Anal.


We show a sufficient criterion to determine if a planar set $\Omega$ is a minimizer of the prescribed curvature functional among all of its subsets. As a special case, we derive a sufficient criterion to determine if $\Omega$ is a self-Cheeger set, i.e. if it minimizes the ratio $P(E)/\textrm{vol}(E)$ among all of its subsets. Specifically, if a Jordan domain $\Omega$ possesses the interior disk property of radius $\textrm{vol}(\Omega)/P(\Omega)$, then it is a self-Cheeger set; if it possesses the strict interior disk property then it is a minimal Cheeger set, i.e. the unique minimizer. As a side effect we provide a way to build self-Cheeger sets.

Keywords: prescribed mean curvature, inner Cheeger formula, Cheeger costant, self-Cheeger sets, perimeter minimizer