Sobolev homeomorphic extensions from two to three dimensions

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 φ: R 2 onto −−→ R 2 in W1,p loc (R 2 , R 2 ) for some p ∈ [1, ∞) admits a homeomorphic extension h: R 3 onto −−→ R 3 in W1,q loc (R 3 , R 3 ) for 1 ⩽ q < 3 2 p. Such an extension result is nearly sharp, as the bound q = 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 quasiconformal extension result.