Strong BV-extension and W1,1-extension domains

Abstract

We show that a bounded domain in a Euclidean space is a W1,1-extension domain if and only if it is a strong BV-extension domain. In the planar case, bounded and strong BV-extension domains are shown to be exactly those BV-extension domains for which the set ∂Ω∖⋃iΩ‾i is purely 1-unrectifiable, where Ωi are the open connected components of R2∖Ω‾.