Homeomorphic Sobolev extensions of parametrizations of Jordan curves

TL;DR

This work identifies a sharp geometric criterion for when boundary parametrizations of Jordan curves admit Sobolev homeomorphic extensions with finite first-order energy. The main result shows that if the Jordan domain $\Omega$ satisfies $\int_{\Omega} (h_{\Omega}(z,z_0))^q \,dz < \infty$ for some $q>1$, then every boundary homeomorphism $\varphi:\partial\mathbb{D}\to\partial\Omega$ extends to a homeomorphism $\Phi\in W^{1,p}(\mathbb{D},\mathbb{C})$ for all $p\in[1,2)$, with the proof leveraging the Koski–Onninen extension theorem via dyadic crosscuts and Koebe distortion. This criterion is shown to be optimal by constructing a Jordan domain (via a Smith–Volterra–Cantor-based tree core fattened into fingers) for which a $W^{1,1}$-extension fails despite finite $L^1$ hyperbolic distance, thus clarifying the boundary data that can arise from finite-energy deformations. The results illuminate the precise boundary regularity requirements for feasible finite-energy deformations in geometric function theory and nonlinear elasticity.

Abstract

Each homeomorphic parametrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is a natural question to ask if such a homeomorphism can be chosen so as to have some Sobolev regularity. This prompts the simplified question: for a homeomorphic embedding of the unit circle into the plane, when can we find a homeomorphism from the unit disk that has the same boundary values and integrable first-order distributional derivatives? We give the optimal geometric criterion for the interior Jordan domain so that there exists a Sobolev homeomorphic extension for any homeomorphic parametrization of the Jordan curve. The problem is partially motivated by trying to understand which boundary values can correspond to deformations of finite energy.

Figures (1)

• Figure 1: Schematic picture of the Jordan domain

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Definition 2.5
• proof : Proof of Theorem \ref{['The1']}