Differentiability almost everywhere of weak limits of bi-Sobolev homeomorphisms

TL;DR

This paper proves that weak limits of bi-Sobolev homeomorphisms with positive Jacobians are differentiable almost everywhere, extending understanding of regularity in nonlinear elasticity and geometric analysis.

Contribution

It establishes the almost everywhere differentiability of weak limits of bi-Sobolev homeomorphisms with positive Jacobians, a novel result in the analysis of nonlinear mappings.

Findings

Weak limits of bi-Sobolev homeomorphisms are differentiable almost everywhere.

Inverse mappings of these limits are also differentiable almost everywhere.

The composition of the limit and its inverse is the identity almost everywhere.

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.