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arXiv · 2207.02044

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

Marco Caroccia·Giorgio Saracco
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Abstract

Given an open, bounded, planar set $Ω$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,|Ω|])$ associating to each $p$ the set of volumes of $p$-Cheeger sets. We show that whenever $Ω$ satisfies some geometric structural assumptions (convex sets are encompassed), the map is injective, and continuous in terms of $Γ$-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.

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Marco Caroccia, Giorgio Saracco. 2023-02-10. Isoperimetric sets and $p$-Cheeger sets are in bijection. https://doi.org/10.1007/s12220-022-01157-x

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