Isoperimetric Sets and p-Cheeger Sets are in Bijection

Abstract

Given an open, bounded, planar set $$\Omega $$ Ω , we consider its p-Cheeger sets and its isoperimetric sets. We study the set-valued map $$\mathfrak {V}:[1/2,+\infty )\rightarrow \mathcal {P}((0,|\Omega |])$$ V : [ 1 / 2 , + ∞ ) → P ( ( 0 , | Ω | ] ) associating to each p the set of volumes of p-Cheeger sets. We show that whenever $$\Omega $$ Ω satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of $$\Gamma $$ Γ -convergence. Moreover, when restricted to $$(1/2, 1)$$ ( 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.