On smooth interior approximation of Sets of Finite Perimeter

Abstract

In this paper, we prove that for any bounded set of finite perimeter $\Omega \subset {\mathbb{R}}^n$, we can choose smooth sets $E_k⋐\Omega$ such that $E_k\to \Omega$ in $L^1$ and \begin{align} \label{moregeneralapproximation} \limsup_{i \rightarrow \infty} P(E_i) \le P(Ω)+C_1(n) \mathscr{H}^{n-1}(\partial Ω\cap Ω^1). \end{align}In the above ${\Omega }^1$ is the measure-theoretic interior of $\Omega$, $P(\cdot )$ denotes the perimeter functional on sets, and $C_1(n)$ is a dimensional constant. Conversely, we prove that for any sets $E_k⋐\Omega$ satisfying $E_k\to \Omega$ in $L^1$, there exists a dimensional constant $C_2(n)$ such that the following inequality holds: \begin{align} \label{gap} \liminf_{k \rightarrow \infty} P(E_k) \ge P(Ω)+ C_2(n) \mathscr{H}^{n-1}(\partial Ω\cap Ω^1). \end{align} In particular, these results imply that for a bounded set $\Omega$ of finite perimeter,\begin{align} \label{char*} \mathscr{H}^{n-1}(\partial Ω\cap Ω^1)=0 \end{align} holds if and only if there exists a sequence of smooth sets $E_k$ such that $E_k⋐\Omega$, $E_k\to \Omega$ in $L^1$ and $P(E_k)\to P(\Omega )$.

This paper was accepted in 04/21/2021 by Proc. AMS, but until now it was still not online. Since a few people have consulted our results, we post the paper on arXiv