On the approximation of finite perimeter sets

Alessandro Carbotti; Simone Cito; Domenico Angelo La Manna; Aldo Pratelli; Giorgio Stefani

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arXiv · 2603.18984

On the approximation of finite perimeter sets

Alessandro Carbotti·Simone Cito·Domenico Angelo La Manna·Aldo Pratelli·Giorgio Stefani

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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.

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BibTeX RIS

Alessandro Carbotti, Simone Cito, Domenico Angelo La Manna, Aldo Pratelli, Giorgio Stefani. 2026-03-19. On the approximation of finite perimeter sets. https://arxiv.org/abs/2603.18984

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