  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-Schoenflies theorem may admit no solution - it is possible to have a
boundary homeomorphism which admits a continuous $W^{1,2}$-extension but not
even a homeomorphic $W^{1,1}$-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.
