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A geometric characterization of planar Sobolev extension domains

Pekka Koskela, Tapio Rajala, Yi Ru-Ya Zhang

Abstract

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

A geometric characterization of planar Sobolev extension domains

Abstract

We characterize bounded simply-connected planar -extension domains for as those bounded domains for which any two points can be connected with a curve satisfying Combined with known results, we obtain the following duality result: a Jordan domain is a -extension domain, , if and only if the complementary domain is a -extension domain.

Paper Structure

This paper contains 23 sections, 59 theorems, 521 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Idea of the proof
  3. Preliminaries
  4. Curves and integrals over curves
  5. Curve condition
  6. Hyperbolic metric
  7. Whitney-type set
  8. Conformal capacity
  9. John domains
  10. Conformal geometry of the exterior domain
  11. Proof of necessity
  12. Necessity in the Jordan case
  13. Inner extension
  14. Proof in the general case
  15. Proof of sufficiency
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $1 < p < 2$ and let $\Omega \subset \mathbb{R}^2$ be a bounded simply connected domain. Then $\Omega$ is a $W^{1,p}$-extension domain if and only if for all $z_1,z_2 \in \mathbb{R}^2 \setminus \Omega$ there exists a curve $\gamma \subset \mathbb{R}^2 \setminus \Omega$ joining $z_1$ and $z_2$ suc

Figures (11)

  • Figure 1: An illustration of the annular parts $\varphi^{-1}(\Gamma_k)$ and $\varphi^{-1}(\gamma_k)$, for $k = 0$, that are considered in Lemma \ref{['lengthtransfer']}.
  • Figure 2: The function $\phi$ is seen to have large value in $\Omega_1$ by observing that any curve $\gamma(w,P_2)$ connecting a point $w \in \Omega_1$ to $P_2$ in $\Omega$ must intersect $\gamma_0 = \gamma_1\cup\gamma_2$. In order to see that $\phi$ has small value near $P_2$ one observes that $\phi$ near $x \in P_2$ can be estimated by integrating $\frac{1}{R_x}$ along a curve with length at most $r_x \le \epsilon R_x$.
  • Figure 3: The curve $\gamma$ is obtained as the image of the curve $\alpha$ under the conformal map $\widetilde{\varphi} \colon \mathbb{R}^2 \setminus \overline {\mathbb{D}} \to \mathbb{R}^2 \setminus \overline {\Omega}$. In the illustration the Whitney squares in $\widetilde{W}_\gamma$ are highlighted.
  • Figure 4: The curve constructed in Theorem \ref{['neceJordan']} can be modified so as to travel inside $\widetilde{\Omega}$ by perturbing slightly the starting point and the endpoint of the intermediate curve $\widetilde{\varphi}(\alpha)$ and by disregarding the unnecessary parts of the concatenated curves. On the left we have the case where the selected points $z_3$ and $z_4$ differ, and on the right the case where they agree.
  • Figure 5: In the inner extension the annular region $\widetilde{\Omega}_\epsilon$ is divided into Whitney-type sets that are obtained by mapping a Whitney-type decomposition of the annulus inside the disk conformally. For the inner part $\Omega_\epsilon$ we use a standard Whitney decomposition. Two pairs of sets $(\widetilde{S}_i,S_i)$ and $(\widetilde{S}_j,S_j)$ are highlighted.
  • ...and 6 more figures

Theorems & Definitions (109)

  • Theorem 1.1
  • Theorem 1.2: Shvartsman
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 99 more