Weak limits of Sobolev homeomorphisms are one to one

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 $p>\lfloor \frac{n}{2}\rfloor$ for $n\ge 4$ or $p\ge 1$ 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.