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Characterizing Sobolev Homeomorphic Extensions via Internal Distances

Aleksis Koski, Jani Onninen, Haiqing Xu

TL;DR

The paper provides a complete characterization of when a boundary embedding $\varphi: \partial\mathbb{D} \to \partial\mathbb{Y}$ admits a Sobolev homeomorphic extension to the disk, via the internal $p$-Douglas condition: for $1<p<\infty$, such an extension exists if and only if $\iint_{\partial\mathbb{D}}\iint_{\partial\mathbb{D}} \frac{[d_{\mathbb{Y}}(\varphi(x),\varphi(y))]^p}{|x-y|^p}\,dx\,dy<\infty$, where $d_{\mathbb{Y}}$ is the internal distance in $\overline{\mathbb{Y}}$. The sufficiency proof uses a novel dyadic geometric extension built from hyperbolic geodesics and dyadic ceilings, while necessity relies on maximal-operator bounds for the spherical maximal function to control the trace behavior. The results yield a Sobolev analogue of the Jordan–Schönflies theorem, with corollaries for Lipschitz and quasiconvex targets, and reveal sharp distinctions at $p=1$, $p>2$, and for rectifiable vs piecewise-smooth boundaries, informing well-posedness in nonlinear elasticity and geometric function theory.

Abstract

We give a full characterization of embeddings of the unit circle that admit a Sobolev homeomorphic extension to the unit disk. As a direct corollary, we establish that for quasiconvex target domains $\mathbb Y$, any homeomorphism $\varphi \colon \partial \mathbb{D} \to \partial \mathbb Y$ that admits a continuous $W^{1,p}$-extension to the unit disk $\mathbb{D}$ also admits a $W^{1,p}$-homeomorphic extension. These Sobolev variants of the classical Jordan-Schönflies theorem are essential for ensuring the well-posedness of variational problems arising in Nonlinear Elasticity and Geometric Function Theory.

Characterizing Sobolev Homeomorphic Extensions via Internal Distances

TL;DR

The paper provides a complete characterization of when a boundary embedding admits a Sobolev homeomorphic extension to the disk, via the internal -Douglas condition: for , such an extension exists if and only if , where is the internal distance in . The sufficiency proof uses a novel dyadic geometric extension built from hyperbolic geodesics and dyadic ceilings, while necessity relies on maximal-operator bounds for the spherical maximal function to control the trace behavior. The results yield a Sobolev analogue of the Jordan–Schönflies theorem, with corollaries for Lipschitz and quasiconvex targets, and reveal sharp distinctions at , , and for rectifiable vs piecewise-smooth boundaries, informing well-posedness in nonlinear elasticity and geometric function theory.

Abstract

We give a full characterization of embeddings of the unit circle that admit a Sobolev homeomorphic extension to the unit disk. As a direct corollary, we establish that for quasiconvex target domains , any homeomorphism that admits a continuous -extension to the unit disk also admits a -homeomorphic extension. These Sobolev variants of the classical Jordan-Schönflies theorem are essential for ensuring the well-posedness of variational problems arising in Nonlinear Elasticity and Geometric Function Theory.

Paper Structure

This paper contains 6 sections, 5 theorems, 49 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}, sufficiency
  3. Maximal operators
  4. Proof of Theorem \ref{['thm:main']}, necessity
  5. Proof of Theorem \ref{['thm:rectifiable_p>2']}
  6. Proof of Theorem \ref{['thm:p=1case']}

Key Result

Theorem 1.1

For $1< p < \infty$, a homeomorphism $\varphi \colon \partial \mathbb D \xrightarrow[]{{}_{\!\!\textnormal{onto\,\,}\!\!}} \partial \mathbb Y$ admits a homeomorphic extension $h \colon \overline{\mathbb D} \xrightarrow[]{{}_{\!\!\textnormal{onto\,\,}\!\!}} \overline{\mathbb Y}$ in the Sobolev class

Figures (10)

  • Figure 1: The intersection of quasihyperbolic geodesics in $\mathbb{D}$ is based on the order of endpoints.
  • Figure 2: Replacing the curves $C_{n,k}$ with the curves $\Gamma_{n,k}$.
  • Figure 3: A step-by-step view of the entire construction on the image side.
  • Figure 4: The construction of the regions $V_{n,k}$ on the target side.
  • Figure 5: The construction of the regions $U_{n,k}$ on the domain side. The boundary here is imagined to be locally flat to simplify the illustration.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • proof
  • proof
  • Remark 5.1
  • proof : Proof of Theorem \ref{['thm:rectifiable_p>2']}
  • proof