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

11 pages

References in corpus (1)

• On the generalized Cheeger problem and an application to 2d strips

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