Characterizing Sobolev Homeomorphic Extensions via Internal Distances

Abstract

We give a full characterization of embeddings of the unit circle that admit a Sobolev homeomorphic extension to the unit disk. As a direct corollary, we establish that for quasiconvex target domains $\mathbb{Y}$, any homeomorphism $\ddot{I}†:\partial \mathbb{D}\to \partial \mathbb{Y}$ that admits a continuous $W^{1,p}$-extension to the unit disk $\mathbb{D}$ also admits a $W^{1,p}$-homeomorphic extension. These Sobolev variants of the classical Jordan-Schönflies theorem are essential for ensuring the well-posedness of variational problems arising in Nonlinear Elasticity and Geometric Function Theory.

10 figures