The Sobolev Jordan-Schönflies problem

Abstract

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