A geometric characterization of planar Sobolev extension domains

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.

arXiv comment82 pages, 11 figures

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