Weighted Cheeger sets are domains of isoperimetry

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 $\mathcal {H}^{n-1}(A^{(1)} \cap \partial A)=0$ satisfies a relative isoperimetric inequality. If $\Omega$ itself is a connected minimizer such that $\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0$, 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 $\mathcal {H}^{n-1}(\Omega ^{(1)} \cap \partial \Omega )=0$.