Calculus of Variations and Geometric Measure Theory
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G. Saracco

A sufficient criterion to determine planar self-Cheeger sets

created by saracco on 28 Jun 2019
modified on 01 Jul 2019

[BibTeX]

Preprint

Inserted: 28 jun 2019
Last Updated: 1 jul 2019

Year: 2019

ArXiv: 1906.12101 PDF

Abstract:

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)/
E
$ among all of its subsets. Specifically, if a Jordan domain $\Omega$ possesses the interior disk property of radius $
\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


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