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

The Sobolev Jordan-Schonflies Problem