Global invertibility of Sobolev mappings with prescribed homeomorphic boundary values

TL;DR

This paper proves that Sobolev mappings with prescribed boundary homeomorphisms are globally invertible and continuous, extending to the boundary as monotone surjections, thus allowing weak interpenetration but preventing folding.

Contribution

It establishes the global invertibility and boundary extension of Sobolev mappings with positive Jacobian and prescribed boundary values, generalizing J.M. Ball's work.

Findings

Mappings extend continuously to boundary

Mappings are monotone surjections

Maps allow weak but not strong interpenetration

Abstract

Let $X, Y \subset \mathbb{R}^n$ be Lipschitz domains, and suppose there is a homeomorphism $\varphi \colon \overline{X} \to \overline{Y}$. We consider the class of Sobolev mappings $f \in W^{1,n} (X, \mathbb{R}^n)$ with a strictly positive Jacobian determinant almost everywhere, whose Sobolev trace coincides with $\varphi$ on $\partial X$. We prove that every mapping in this class extends continuously to $\overline{X}$ and is a monotone (continuous) surjection from $\overline{X}$ onto $\overline{Y}$ in the sense of C.B. Morrey. As monotone mappings, they may squeeze but not fold the reference configuration $X$. This behavior reflects weak interpenetration of matter, as opposed to folding, which corresponds to strong interpenetration. Despite allowing weak interpenetration of matter, these maps are globally invertible, generalizing the pioneering work of J.M. Ball.