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

TL;DR

This paper establishes an equivalence between $W^{1,1}$-extension domains and strong $BV$-extension domains in Euclidean spaces, providing a geometric characterization in the planar case involving $1$-unrectifiability.

Contribution

It proves that bounded Euclidean domains are $W^{1,1}$-extension domains if and only if they are strong $BV$-extension domains, and characterizes planar domains via boundary unrectifiability.

Findings

Equivalence of $W^{1,1}$- and strong $BV$-extension domains in Euclidean spaces.

Geometric characterization of planar $BV$-extension domains.

Boundary structure involving $1$-unrectifiability in the planar case.

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 $\mathbb{R}^2\setminus\overline{\Omega}$.