## 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 W^{1,p}-Sobolev homeomorphic extension is that it first admits a continuous W^{1,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 W^{1,p}-extension but does not even admit a W^{1,1}-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 W_{loc}^{1,p}-homeomorphic extension to C if and only if it admits a W^{1,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 W^{1,2}(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 W^{1,p}-Sobolev extension to D, then it admits a homeomorphic extension to C in W_{loc}^{1,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‾→_{onto}Y‾ with finite Dirichlet energy.

Original language | English (US) |
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Article number | 108795 |

Journal | Advances in Mathematics |

Volume | 413 |

DOIs | |

State | Published - Jan 15 2023 |

## Keywords

- Sobolev extensions
- Sobolev homeomorphisms

## ASJC Scopus subject areas

- General Mathematics