Weak limits of Sobolev homeomorphisms are one to one

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 Ω⊆ Rⁿ be a domain and let p>/2 for n≥ 4 or p≥ 1 for $n=2,3$. Assume that fₖ∈ W1,p is a sequence of homeomorphisms such that fₖ f weakly in W1,p and assume that Jf>0 a.e. Then we show that f is injective a.e.