Homeomorphic Sobolev extensions of parametrizations of Jordan curves

TL;DR

This paper establishes the optimal geometric conditions under which a homeomorphic parametrization of a Jordan curve can be extended to a Sobolev homeomorphism of the entire plane, addressing boundary deformation energy.

Contribution

It provides the first precise geometric criterion for Sobolev extensions of Jordan curve parametrizations, advancing understanding of boundary deformations with finite energy.

Findings

Identifies the optimal geometric condition for Sobolev extensions.

Characterizes when boundary parametrizations admit finite energy deformations.

Advances the theory of Sobolev homeomorphic extensions of Jordan curves.

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.