Strong $BV$-extension and $W^{1,1}$-extension domains

Abstract

We show that a bounded domain in a Euclidean space is a $W^{1,1}$-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 $\partial \Omega \setminus {\bigcup }_i{\overline{\Omega }}_i$ is purely $1$-unrectifiable, where ${\Omega }_i$ are the open connected components of ${\mathbb{R}}^2\setminus \overline{\Omega }$.

Implication (3) to (2) in Theorem 1.3 has been corrected and other minor changes have been done. 31 pages, 2 figures