On indecomposable sets with applications

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. $||1_F-1_{\mathbb{F}}|{|}_{BV({\mathbb{R}}^2)}=0$. 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 $||1_S-1_{{\mathbb{S}}_{\epsilon }}|{|}_{BV({\mathbb{R}}^2)}<\epsilon$. We apply this corollary to give a short proof that locally quasiminimizing sets in the plane are $B{V}_l$ extension domains.

20 pages

References in corpus (1)

• Regularity of sets with quasiminimal boundary surfaces in metric spaces

Cited by in corpus (1)

• A new Federer-type characterization of sets of finite perimeter in metric spaces