On indecomposable sets with applications

• Published in: arXiv.org 2013-10
• Main Author: Lorent, Andrew
• Format: Article
• Language: English
• Subjects: Characteristic functions; Domains
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1305.3264

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)}