Weak limits of Sobolev homeomorphisms are one to one

TL;DR

This paper demonstrates that weak limits of Sobolev homeomorphisms with positive Jacobian are almost everywhere injective, ensuring non-interpenetration of matter in nonlinear elasticity models under minimal assumptions.

Contribution

It proves that weak limits of Sobolev homeomorphisms with positive Jacobian are almost everywhere injective, extending the understanding of deformation limits in nonlinear elasticity.

Findings

Weak limits of Sobolev homeomorphisms are a.e. injective.

Injectivity is preserved under weak convergence with positive Jacobian.

Results apply for Sobolev spaces with minimal regularity assumptions.

Abstract

We prove that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions. Let $\Omega\subseteq \mathbb{R}^n$ be a domain and let $p>\left\lfloor\frac{n}{2}\right\rfloor$ for $n\geq 4$ or $p\geq 1$ for $n=2,3$. Assume that $f_k\in W^{1,p}$ is a sequence of homeomorphisms such that $f_k\rightharpoonup f$ weakly in $W^{1,p}$ and assume that $J_f>0$ a.e. Then we show that $f$ is injective a.e.