# Sobolev homeomorphic extensions

## TL;DR

This paper proves the existence of Sobolev homeomorphic extensions of boundary homeomorphisms between Jordan domains with rectifiable boundaries, and explores conditions for higher integrability based on hyperbolic growth.

## Contribution

It establishes new results on Sobolev homeomorphic extensions in complex domains, including conditions for $W^{1,p}$ regularity and examples showing necessity of assumptions.

## Key findings

- Existence of $W^{1,1}$ Sobolev homeomorphic extensions for boundary homeomorphisms.
- Extensions in $W^{1,p}$ for $p ext{ in } (1,2)$ under hyperbolic growth conditions.
- Examples demonstrating that boundary rectifiability and hyperbolic growth assumptions are necessary.

## Abstract

Let $\mathbb X$ and $\mathbb Y$ be $\ell$-connected Jordan domains, $\ell \in \mathbb N$, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism $\varphi \colon \partial \mathbb X \to \partial \mathbb Y$ admits a Sobolev homeomorphic extension $h \colon \overline{\mathbb X} \to \overline{\mathbb Y}$ in $W^{1,1} (\mathbb X, \mathbb C)$. If instead $\mathbb X$ has $s$-hyperbolic growth with $s>p-1$, we show the existence of such an extension lies in the Sobolev class $W^{1,p} (\mathbb X, \mathbb C)$ for $p\in (1,2)$. Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of $W^{1,2}$-homeomorphic extensions subject to a given boundary data.