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 $E_k \Subset \Omega$ such that $E_k \rightarrow \Omega$ in $L^1$ and \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 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 $E_k \Subset \Omega$ satisfying $E_k \rightarrow \Omega$ in $L^1$, there exists a dimensional constant $C_2(n)$ such that the following inequality holds: \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 set $\Omega$ of finite perimeter, \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 sets $E_k$ such that $E_k \Subset \Omega$, $E_k \rightarrow \Omega$ in $L^1$ and $P(E_k) \rightarrow P(\Omega )$.