Isoperimetric sets and $p$-Cheeger sets are in bijection

Abstract

Given an open, bounded, planar set $\Omega$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac{1}{2},+\infty )\to \mathcal{P}((0,|\Omega |])$ associating to each $p$ the set of volumes of $p$-Cheeger sets. We show that whenever $\Omega$ satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of $\Gamma$-convergence. Moreover, when restricted to $(\frac{1}{2},1)$ such a map is univalued and is in bijection with its image. As a consequence of our analysis we derive some fine boundary regularity result.

References in corpus (5)

• The Cheeger constant of a Jordan domain without necks
• On the higher Cheeger problem
• Minimizers of the prescribed curvature functional in a Jordan domain with no necks
• On the Cheeger problem for rotationally invariant domains
• The Cheeger constant of curved tubes