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
Abstract:
Given an open, bounded, planar set $\Omega$, we consider its $p$-Cheeger sets and its isoperimetric sets. We study the set-valued map $\mathfrak{V}:[\frac12,+\infty)\rightarrow\mathcal{P}((0,
\Omega
])$ 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 $(\frac 12, 1)$ 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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