On smooth interior approximation of sets of finite perimeter

Abstract

In this paper, we prove that for any bounded set of finite perimeterΩ⊂Rn\Omega \subset \mathbb {R}^n, we can choose smooth setsEk⋐ΩE_k \Subset \Omegasuch thatEk→ΩE_k \rightarrow \OmegainL1L^1andlim supi→∞P(Ei)≤P(Ω)+C1(n)Hn−1(∂Ω∩Ω1).\begin{align*} \limsup _{i \rightarrow \infty } P(E_i) \le P(\Omega )+C_1(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*}In the aboveΩ1\Omega ^1is the measure-theoretic interior ofΩ\Omega,P(⋅)P(\cdot )denotes the perimeter functional on sets, andC1(n)C_1(n)is a dimensional constant.

Conversely, we prove that for any setsEk⋐ΩE_k \Subset \OmegasatisfyingEk→ΩE_k \rightarrow \OmegainL1L^1, there exists a dimensional constantC2(n)C_2(n)such that the following inequality holds:lim infk→∞P(Ek)≥P(Ω)+C2(n)Hn−1(∂Ω∩Ω1).\begin{align*} \liminf _{k \rightarrow \infty } P(E_k) \ge P(\Omega )+ C_2(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*}In particular, these results imply that for a bounded setΩ\Omegaof finite perimeter,Hn−1(∂Ω∩Ω1)=0\begin{align*} \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1)=0 \end{align*}holds if and only if there exists a sequence of smooth setsEkE_ksuch thatEk⋐ΩE_k \Subset \Omega,Ek→ΩE_k \rightarrow \OmegainL1L^1andP(Ek)→P(Ω)P(E_k) \rightarrow P(\Omega ).