On smooth interior approximation of sets of finite perimeter

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 ).