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On the approximation of finite perimeter sets

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

Abstract

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

On the approximation of finite perimeter sets

Abstract

We prove that if is a set with finite perimeter with , then any set of finite perimeter can be approximated by a polyhedral or smooth bounded set in such a way that both the total perimeter of and the perimeter of inside are approximated by those of , and the boundary of has negligible intersection with the boundary of . 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 with a set which agrees with and has the same boundary inside , while sharing no common boundary with , and does so without substantially altering the perimeter or the volume of the original set.
Paper Structure (7 sections, 14 theorems, 99 equations, 4 figures)

This paper contains 7 sections, 14 theorems, 99 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main technical result, Theorem A
  4. Proof of Theorem B and Theorem C
  5. The proof of Theorem B
  6. The proof of Theorem C
  7. Sharpness of the assumptions

Key Result

Theorem 1.1

Let $\Omega\subseteq\mathbb R^N$ be a bounded open set with Lipschitz boundary, and let $E\subseteq\mathbb R^N$ be a set of finite perimeter in $\Omega$. Then, for every $\varepsilon>0$ there exists a polyhedral set $F$ such that

Figures (4)

  • Figure 1: The green set is $E$, the dashed set is $F$ given by Theorem A.
  • Figure 2: Definition of $\Gamma^\pm_E$.
  • Figure 3: The green set is $E$, the dashed set is $\widetilde{E}$, the thicker black line is $\partial^*\Omega$.
  • Figure 4: The situation in Example \ref{['<<>>']}; the density $g$ is very low near the boundary of $E$, but very large in the orange region.

Theorems & Definitions (33)

  • Theorem 1.1: Approximation with a Lipschitz set $\Omega$
  • Definition 1.2
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 23 more