Sobolev homeomorphic extensions

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 $\phi :\partial \mathbb{X}\to \partial \mathbb{Y}$ admits a Sobolev homeomorphic extension $h:\overline{\mathbb{X}}\to \overline{\mathbb{Y}}$ 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.

25 pages, 5 figures