Sobolev homeomorphic extensions from two to three dimensions

Abstract

We study the basic question of characterizing which boundary homeomorphisms of the unit sphere can be extended to a Sobolev homeomorphism of the interior in 3D space. While the planar variants of this problem are well-understood, completely new and direct ways of constructing an extension are required in 3D. We prove, among other things, that a Sobolev homeomorphism $\phi :{\mathbb{R}}^2\to {\mathbb{R}}^2$ in $W_{loc}^{1,p}({\mathbb{R}}^2,{\mathbb{R}}^2)$ for some $p\in [1,\infty )$ admits a homeomorphic extension $h:{\mathbb{R}}^3\to {\mathbb{R}}^3$ in $W_{loc}^{1,q}({\mathbb{R}}^3,{\mathbb{R}}^3)$ for $1\le q<\frac{3}{2}p$. Such an extension result is nearly sharp, as the bound $q=\frac{3}{2}p$ cannot be improved due to the Hölder embedding. The case $q=3$ gains an additional interest as it also provides an $L^1$-variant of the celebrated Beurling-Ahlfors extension result.

47 pages, 15 figures