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

TL;DR

This paper establishes a bijection between isoperimetric and $p$-Cheeger sets in certain planar domains, revealing structural and regularity properties of these geometric sets across a range of $p$ values.

Contribution

It proves the bijection and continuity of the volume map for $p$-Cheeger sets under geometric assumptions, extending understanding of their structure and regularity.

Findings

The volume map is injective and continuous under structural assumptions.

The map is bijective on $(rac{1}{2},1)$, establishing a one-to-one correspondence.

Derived boundary regularity results for $p$-Cheeger sets.

Abstract

Given an open, bounded, planar set $\Omega$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,|\Omega|])$ 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 $(\frac 12, 1)$ 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.