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 $\phi :\partial \mathbb{X}\stackrel{{}_{\text{onto}}}{\to }\partial \mathbb{Y}$ admits a Sobolev homeomorphic extension $h:\overline{\mathbb{X}}\stackrel{{}_{\text{onto}}}{\to }\overline{\mathbb{Y}}$ in ${\mathcal{W}}^{1,1}(\mathbb{X},\mathbb{C})$. If instead $\mathbb{X}$ has $s$-hyperbolic growth with $s>p-1$, we show the existence of such an extension in the Sobolev class ${\mathcal{W}}^{1,p}(\mathbb{X},\mathbb{C})$ 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 ${\mathcal{W}}^{1,2}$-homeomorphic extensions with given boundary data.