Calculus of Variations and Geometric Measure Theory

M. Caroccia - G. Saracco

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

created by caroccia on 05 Jul 2022
modified by saracco on 10 Feb 2023


Published Paper

Inserted: 5 jul 2022
Last Updated: 10 feb 2023

Journal: J. Geom. Anal.
Volume: 33
Number: 129
Year: 2023
Doi: 10.1007/s12220-022-01157-x

ArXiv: 2207.02044 PDF


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