Calculus of Variations and Geometric Measure Theory

G. Saracco

Weighted Cheeger sets are domains of isoperimetry

created by saracco on 09 Oct 2016
modified on 09 Jun 2018


Published Paper

Inserted: 9 oct 2016
Last Updated: 9 jun 2018

Journal: Manuscripta Math.
Volume: 156
Number: 3--4
Pages: 371--381
Year: 2018
Doi: 10.1007/s00229-017-0974-z

ArXiv: 1610.02717 PDF


We consider a generalization of the Cheeger problem in a bounded, open set $\Omega$ by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any connected minimizer $A$ of this weighted Cheeger problem such that $H^{n-1}(A^{(1)} \cap \partial A)=0$ satisfies a relative isoperimetric inequality. If $\Omega$ itself is a connected minimizer such that $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$, then it allows the classical Sobolev and $BV$ embeddings and the classical $BV$ trace theorem. The same result holds for any connected minimizer whenever the weights grant the regularity of perimeter-minimizer sets and $\Omega$ is such that $
\partial \Omega
=0$ and $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$.

Keywords: Cheeger problem, Sobolev embeddings, trace theorems