Sobolev homeomorphic extensions onto John domains

摘要

Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical Jordan-Schiienflies theorem may admit no solution - it is possible to have a boundary homeomorphism which admits a continuous W-1,W-2-extension but not even a homeomorphic W-1,W-2-extension. We prove that if the target is assumed to be a John disk, then any boundary homeomorphism from the unit circle admits a Sobolev homeomorphic extension for all exponents p < 2. John disks, being one sided quasidisks, are of fundamental importance in Geometric Function Theory. (C) 2020 Elsevier Inc. All rights reserved.