The Sobolev Jordan-Schönflies problem

We consider the planar unit disk D as reference configuration and a Jordan domain Y deformed configuration, study problem of extending given boundary homeomorphism ?:?D?onto?Y Sobolev complex plane. Investigating such variant classical Jordan-Schönflies theorem is motivated by well-posedness related pure displacement variational questions in theory Nonlinear Elasticity (NE) Geometric Function Theory (GFT). Clearly, necessary condition for mapping ? to admit W1,p-Sobolev homeomorphic extension that it first admits continuous extension. For an arbitrary target this, however, not sufficient. Indeed, each p<? we construct which W1,p-extension but does even W1,1-homeomorphic Second, quasidisk whole range p, prove Wloc1,p-homeomorphic C if only disk. Quasidisks have been subject intensive GFT. They do allow singularities on cusps. Third, any power-type cusp there from circle whose harmonic has finite Dirichlet energy W1,2(D,C). Surprisingly, integral (p=2) plays unique role Problem case targets. Even more, fourth piecewise smooth boundary, p?2 D, then Wloc1,p(C,C). Fifth, addition quasiconvex, one-sided solution when p=2. show h:D??ontoY? with energy.