Calculus of Variations and Geometric Measure Theory
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G. Saracco

Weighted Cheeger sets are domains of isoperimetry

created by saracco on 09 Oct 2016
modified on 12 Sep 2017


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Inserted: 9 oct 2016
Last Updated: 12 sep 2017

Journal: Manuscripta Math.
Pages: 1-11
Year: 2017
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


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