On indecomposable sets with applications

TL;DR

This paper investigates the properties of indecomposable sets in the plane, showing their characteristic functions are BV equivalent to closed sets, and explores approximation and extension properties with implications for geometric measure theory.

Contribution

It establishes BV equivalence of indecomposable sets to closed sets in the plane and introduces approximation techniques for sets of finite perimeter.

Findings

Characteristic functions of indecomposable sets in the plane are BV equivalent to closed sets.

In higher dimensions, this BV equivalence does not hold.

Sets of finite perimeter can be approximated by unions of indecomposable components with controlled boundary differences.

Abstract

In this note we show the characteristic function of every indecomposable set $F$ in the plane is $BV$ equivalent to the characteristic function a closed set $\mathbb{F}$, i.e. $||\mathbb{1}_{F}-\mathbb{1}_{\mathbb{F}}||_{BV(\mathbb{R}^2)}=0$. We show by example this is false in dimension three and above. As a corollary to this result we show that for every $\epsilon>0$ a set of finite perimeter $S$ can be approximated by a closed subset $\mathbb{S}_{\epsilon}$ with finitely many indecomposable components and with the property that $H^1(\partial^M \mathbb{S}_{\epsilon}\backslash \partial^M S)=0$ and $||\mathbb{1}_{S}-\mathbb{1}_{\mathbb{S}_{\epsilon}}||_{BV(\mathbb{R}^2)}<\epsilon$. We apply this corollary to give a short proof that locally quasiminimizing sets in the plane are $BV_l$ extension domains.