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

DOI10.1007/S12220-022-01157-XarXiv2207.02044MaRDI QIDQ6404049

Giorgio Saracco, M. Caroccia

Publication date: 5 July 2022

Abstract: Given an open, bounded, planar set $Omega$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $mathfrakV:[frac12,+infty)ightarrowmathcalP((0,|Omega|])$ 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 $(frac12,1)$ 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.

Mathematics Subject Classification ID

Estimates of eigenvalues in context of PDEs (35P15) Minimal surfaces in differential geometry, surfaces with prescribed mean curvature (53A10) Optimization of shapes other than minimal surfaces (49Q10)

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