On the approximation of finite perimeter sets

Abstract

We prove that if $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^N$ is a set with finite perimeter with ${\mathcal{H}}^{N-1}(\partial \hat{I}\text{c◯}\setminus {\partial }^{\ast }\hat{I}\text{c◯})=0$, then any set of finite perimeter $E\subseteq {\mathbb{R}}^N$ can be approximated by a polyhedral or smooth bounded set $F$ in such a way that both the total perimeter of $E$ and the perimeter of $E$ inside $\hat{I}\text{c◯}$ are approximated by those of $F$, and the boundary of $F$ has negligible intersection with the boundary of $\hat{I}\text{c◯}$. In addition, we address the approximation for perimeter and volume with densities, and we present counterexamples illustrating the sharpness of our assumptions. Our constructions rely on a technical result that replaces $E$ with a set $F$ which agrees with $E$ and has the same boundary inside $\hat{I}\text{c◯}$, while sharing no common boundary with $\hat{I}\text{c◯}$, and does so without substantially altering the perimeter or the volume of the original set.

26 pages, 4 figures