论文标题
等级套件和$ p $ -Cheeger套件是在两者中进行的
Isoperimetric sets and $p$-Cheeger sets are in bijection
论文作者
论文摘要
给定一个开放的，有限的平面套件$ω$，我们考虑其$ p $ -Cheeger套件及其等值套件。我们研究set-valued地图$ \ mathfrak {v}：[\ frac12，+\ infty）\ rightarrow \ mathcal \ mathcal {p}（（（0，|ω|]）$与每个$ p $关联的$ p $ $ p $ -Cheeger设置的集合。此外，该地图是$γ$ - convergence的连续的，当时仅限于$（\ frac 12，1）$ blive of the Image blive。
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.
