Differentiability almost everywhere of weak limits of bi-Sobolev homeomorphisms

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 ${\sup }_k(\mathrm{\parallel }{f}_k{\mathrm{\parallel }}_{W^{1,n-1}}+\mathrm{\parallel }{f}_k^{-1}{\mathrm{\parallel }}_{W^{1,p}})<\mathrm{\infty }$. 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.