Global invertibility for orientation-preserving Sobolev maps via invertibility on or near the boundary

TL;DR

This paper extends classical results on the invertibility of Sobolev maps by showing that boundary invertibility or approximation suffices for global invertibility, with applications in nonlinear elasticity.

Contribution

It demonstrates that boundary invertibility or approximation alone ensures global invertibility of Sobolev maps, avoiding the need for homeomorphic extension.

Findings

Global invertibility follows from boundary invertibility or approximation.

The approach applies to any dimension and avoids homeomorphic extension.

Applications include existence of boundary-value free homeomorphic minimizers in elasticity.

Abstract

By a result of John Ball (1981), a locally orientation preserving Sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension. As shown here for any dimension, the conclusions of Ball's theorem and related results can be reached while completely avoiding the problem of homeomorphic extension. For suitable domains, it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps. An application in Nonlinear Elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed. As a tool in the proofs, strictly orientation-preserving maps and their global invertibility properties are studied from a purely topological point of view.