Planar $W^{1,\,1}$-extension domains

TL;DR

This paper characterizes planar $W^{1,1}$-extension domains using geometric curve conditions and establishes equivalences between $W^{1,1}$, $BV$, and $W^{1,\, ext{infinity}}$ extension properties for Jordan domains.

Contribution

It provides a geometric criterion for planar $W^{1,1}$-extension domains and links these to $BV$ and $W^{1,\, ext{infinity}}$ extension properties.

Findings

A bounded simply connected planar domain is a $W^{1,1}$-extension domain iff certain curve integrals are bounded.

Planar Jordan domains are $W^{1,1}$-extension iff they are $BV$-extension domains.

The complement of a Jordan domain being a $W^{1,\infty}$-extension domain is equivalent to the domain being a $W^{1,1}$-extension domain.

Abstract

We show that a bounded planar simply connected domain $\Omega$ is a $W^{1,\,1}$-extension domain if and only if for every pair $x,y$ of points in $\Omega^c$ there exists a curve $\gamma \subset \Omega^c$ connecting $x$ and $y$ with $$ \int_\gamma \frac{1}{\chi_{\mathbb R^2\setminus \partial\Omega}(z)}\,ds(z) \le C|x-y|.$$ Consequently, a planar Jordan domain $\Omega$ is a $W^{1,\,1}$-extension domain if and only if it is a $BV$-extension domain, and if and only if its complementary domain $\tilde \Omega$ is a $W^{1,\,\infty}$-extension domain.