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:R2→ontoR2 in Wloc1,p(R2,R2) for some p∈[1,∞) admits a homeomorphic extension h:R3→ontoR3 in Wloc1,q(R3,R3) for 1⩽q<32p. Such an extension result is nearly sharp, as the bound q=32p cannot be improved due to the Hölder embedding. The case q=3 gains an additional interest as it also provides an L1-variant of the celebrated Beurling-Ahlfors quasiconformal extension result.