Zero volume boundary for extension domains from Sobolev to BV

Abstract

In this note, we prove that the boundary of a \((W^{1, p}, BV)\)-extension domain is of volume zero under the assumption that the domain \({\Omega }\) is 1-fat at almost every \(x\in \partial {\Omega }\). Especially, the boundary of any planar \((W^{1, p}, BV)\)-extension domain is of volume zero.