  We consider the planar unit disk $\mathbb D$ as the reference configuration
and a Jordan domain $\mathbb Y$ as the deformed configuration, and study the
problem of extending a given boundary homeomorphism $\varphi \colon \partial
\mathbb D \to \partial \mathbb Y$ as a Sobolev homeomorphism of the complex
plane. Investigating such a Sobolev variant of the classical
Jordan-Sch\"onflies theorem is motivated by the well-posedness of the related
pure displacement variational questions in the theory of Nonlinear Elasticity
(NE) and Geometric Function Theory (GFT). Clearly, the necessary condition for
the boundary mapping $\varphi$ to admit a $W^{1,p}$-Sobolev homeomorphic
extension is that it first admits a continuous $W^{1,p}$-Sobolev extension. For
an arbitrary target domain $\mathbb Y$ this, however, is not sufficent.
