Given an open, bounded, planar set Ω , we consider its p-Cheeger sets and its isoperimetric sets. We study the set-valued map V:[1/2,+∞)→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 (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.
Isoperimetric Sets and p-Cheeger Sets are in Bijection / Caroccia, Marco; Saracco, Giorgio. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - 33:(2023), pp. 12901-12917. [10.1007/s12220-022-01157-x]
Isoperimetric Sets and p-Cheeger Sets are in Bijection
Saracco, Giorgio
2023-01-01
Abstract
Given an open, bounded, planar set Ω , we consider its p-Cheeger sets and its isoperimetric sets. We study the set-valued map V:[1/2,+∞)→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 (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.File | Dimensione | Formato | |
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