On indecomposable sets with applications

Mathematics Department, University of Cincinnati, 2600 Clifton Ave. Cincinnati OH 45221.
lorentaw@uc.edu

Received: 14 May 2013
Revised: 29 October 2013

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 . We show by example this is false in dimension three and above. As a corollary to this result we show that for every ϵ > 0 a set of finite perimeter S can be approximated by a closed subset with finitely many indecomposable components and with the property that and . We apply this corollary to give a short proof that locally quasiminimizing sets in the plane are BV_l extension domains.