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Weak limits of Sobolev homeomorphisms are one to one (2409.01260v1)
Published 2 Sep 2024 in math.FA
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.