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Differentiability almost everywhere of weak limits of bi-Sobolev homeomorphisms (2302.07578v1)
Published 15 Feb 2023 in math.FA
Abstract: 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.