Weak limits of Sobolev homeomorphisms are one to one almost everywhere
Ondrěj Bouchala, Stanislav Hencl, Zheng Zhu · Calculus of Variations and Partial Differential Equations · 2025
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 $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n be a domain and let $$p>\left\lfloor \frac{n}{2}\right\rfloor $$ p > n 2 for $$n\ge 4$$ n ≥ 4 or $$p\ge 1$$ p ≥ 1 for $$n=2,3$$ n = 2 , 3 . Assume that $$f_k\in W^{1,p}$$ f k ∈ W 1 , p is a sequence of homeomorphisms such that $$f_k\rightharpoonup f$$ f k ⇀ f weakly in $$W^{1,p}$$ W 1 , p and assume that $$J_f>0$$ J f > 0 a.e. Then we show that f is injective a.e.