On the approximation of finite perimeter sets

TL;DR

This paper proves that sets of finite perimeter can be approximated by smooth or polyhedral sets while preserving perimeter properties, addressing approximation with densities and providing counterexamples for sharpness.

Contribution

It introduces a method to approximate finite perimeter sets with smooth or polyhedral sets, maintaining perimeter and volume properties, and explores approximation with densities.

Findings

Sets with finite perimeter can be approximated by smooth or polyhedral sets.

The approximation preserves perimeter and volume within a domain.

Counterexamples demonstrate the sharpness of the assumptions.

Abstract

We prove that if $\Omega\subseteq\mathbb{R}^N$ is a set with finite perimeter with $\mathscr{H}^{N-1}(\partial \Omega\setminus\partial^* \Omega)=0$, 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 $\Omega$ are approximated by those of $F$, and the boundary of $F$ has negligible intersection with the boundary of $\Omega$. 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 $\Omega$, while sharing no common boundary with $\Omega$, and does so without substantially altering the perimeter or the volume of the original set.