The Sobolev Jordan-Schonflies Problem

TL;DR

This paper investigates the conditions under which boundary homeomorphisms of the disk extend as Sobolev homeomorphisms to the plane, with implications for nonlinear elasticity and geometric function theory.

Contribution

It analyzes Sobolev extension problems for boundary maps in the context of the Sobolev Jordan-Schönflies theorem, highlighting conditions for existence of Sobolev homeomorphic extensions.

Findings

Necessary condition: boundary map admits a continuous Sobolev extension.

Not all boundary maps admit Sobolev homeomorphic extensions for arbitrary targets.

Implications for nonlinear elasticity and geometric function theory.

Abstract

We consider the planar unit disk $\mathbb D$ as the reference configuration and a Jordan domain $\mathbb Y$ as the deformed configuration, and study the problem of extending a given boundary homeomorphism $\varphi \colon \partial \mathbb D \to \partial \mathbb Y$ as a Sobolev homeomorphism of the complex plane. Investigating such a Sobolev variant of the classical Jordan-Sch\"onflies theorem is motivated by the well-posedness of the related pure displacement variational questions in the theory of Nonlinear Elasticity (NE) and Geometric Function Theory (GFT). Clearly, the necessary condition for the boundary mapping $\varphi$ to admit a $W^{1,p}$-Sobolev homeomorphic extension is that it first admits a continuous $W^{1,p}$-Sobolev extension. For an arbitrary target domain $\mathbb Y$ this, however, is not sufficent.