Weighted Cheeger sets are domains of isoperimetry

TL;DR

This paper generalizes the Cheeger problem by incorporating Finsler-type surface energies and weighted volumes, establishing isoperimetric inequalities for minimizers and linking them to classical Sobolev and BV embeddings.

Contribution

It introduces a weighted Cheeger problem with Finsler-type energies and proves isoperimetric inequalities for its minimizers, extending classical results to more general settings.

Findings

Connected minimizers satisfy relative isoperimetric inequalities.

Weighted Cheeger sets allow classical Sobolev and BV embeddings.

Results hold under regularity conditions on weights and boundary measures.

Abstract

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 $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$.