On smooth interior approximation of sets of finite perimeter

Abstract

In this paper, we prove that for any bounded set of finite perimeter Ω ⊂ R n \Omega \subset \mathbb {R}^n , we can choose smooth sets E k ⋐ Ω E_k \Subset \Omega such that E k → Ω E_k \rightarrow \Omega in L 1 L^1 and lim sup i → ∞ P ( E i ) ≤ P ( Ω ) + C 1 ( n ) H n − 1 ( ∂ Ω ∩ Ω 1 ) . \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 Ω 1 \Omega ^1 is the measure-theoretic interior of Ω \Omega , P ( ⋅ ) P(\cdot ) denotes the perimeter functional on sets, and C 1 ( n ) C_1(n) is a dimensional constant. Conversely, we prove that for any sets E k ⋐ Ω E_k \Subset \Omega satisfying E k → Ω E_k \rightarrow \Omega in L 1 L^1 , there exists a dimensional constant C 2 ( n ) C_2(n) such that the following inequality holds: lim inf k → ∞ P ( E k ) ≥ P ( Ω ) + C 2 ( n ) H n − 1 ( ∂ Ω ∩ Ω 1 ) . \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 Ω \Omega of finite perimeter, H n − 1 ( ∂ Ω ∩ Ω 1 ) = 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 sets E k E_k such that E k ⋐ Ω E_k \Subset \Omega , E k → Ω E_k \rightarrow \Omega in L 1 L^1 and P ( E k ) → P ( Ω ) P(E_k) \rightarrow P(\Omega ) .