Global invertibility of Sobolev mappings with prescribed homeomorphic boundary values

Abstract

Let $X, Y \subset \mathbb{R}^n$ be Lipschitz domains, and suppose there is a homeomorphism $φ\colon \overline{X} \to \overline{Y}$. 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 $φ$ 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.