Controlled diffeomorphic extension of homeomorphisms

TL;DR

This paper establishes a precise equivalence between the finite dyadic energy of boundary homeomorphisms and the existence of energy-finite diffeomorphic extensions within certain Jordan domains, advancing understanding of geometric function theory.

Contribution

It proves that boundary homeomorphisms with finite dyadic energy can be extended to finite-energy diffeomorphisms inside the domain, providing a new characterization of such extensions.

Findings

Finite dyadic energy of boundary homeomorphisms is equivalent to existence of finite-energy diffeomorphic extensions.

Characterization of energy conditions for extensions in chord-arc Jordan domains.

Advances in understanding the extension problem for homeomorphisms with energy constraints.

Abstract

Let $\Omega$ be an internal chord-arc Jordan domain and $\varphi:\mathbb S\rightarrow\partial\Omega$ be a homeomorphism. We show that $\varphi$ has finite dyadic energy if and only if $\varphi$ has a diffeomorphic extension $h: \mathbb D\rightarrow \Omega$ which has finite energy.