This paper investigates the differentiability of weak limits of bi-Sobolev homeomorphisms. Given $p>n-1$, consider a sequence of homeomorphisms $f_k$ with positive Jacobians $J_{ f_k} >0$ almost everywhere and $\sup_k(\|f_{ k}\|_{ W^{ 1,n-1}} + \|f_{ k}^{ -1}\|_{ W^{ 1,p}}) <\infty$. We prove that if $f$ and $h$ are weak limits of $f_k$ and $f_k^{ -1}$, respectively, with positive Jacobians $J_f>0$ and $J_h>0$ a.e., then $h(f(x))=x$ and $f(h(y))=y$ both hold a.e.\ and $f$ and $h$ are differentiable almost everywhere.
Categories: math.FA