Zero volume boundary for extension domains from Sobolev to BV

Abstract

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