A geometric characterization of planar Sobolev extension domains

TL;DR

This paper provides a geometric characterization of planar Sobolev extension domains for 1 < p < 2, linking domain geometry with extension properties via curve integrals involving boundary distance.

Contribution

It introduces a new geometric criterion for planar Sobolev extension domains and establishes a duality between the extension properties of a domain and its complement.

Findings

Characterization of planar Sobolev extension domains using curve integrals.

Duality between domain and complement extension properties.

Extension domains are characterized by boundary-distance integrals along connecting curves.

Abstract

We characterize bounded simply-connected planar $W^{1,p}$-extension domains for $1 < p <2$ as those bounded domains $\Omega \subset \mathbb R^2$ for which any two points $z_1,z_2 \in \mathbb R^2 \setminus \Omega$ can be connected with a curve $\gamma\subset \mathbb R^2 \setminus \Omega$ satisfying $$\int_{\gamma} dist(z,\partial \Omega)^{1-p}\, dz \lesssim |z_1-z_2|^{2-p}.$$ Combined with known results, we obtain the following duality result: a Jordan domain $\Omega \subset \mathbb R^2$ is a $W^{1,p}$-extension domain, $1 < p < \infty$, if and only if the complementary domain $\mathbb R^2 \setminus \bar\Omega$ is a $W^{1,p/(p-1)}$-extension domain.