$L^{1}_{loc}$-convergence of Jacobians of Sobolev homeomorphisms via area formula

TL;DR

This paper establishes conditions under which the Jacobians of Sobolev homeomorphisms converge in local L^1, using Federer's area formula, for sequences converging in Sobolev spaces with specific integrability conditions.

Contribution

It proves L^1_{loc} convergence of Jacobians for Sobolev homeomorphisms under certain convergence and measure-preserving conditions, extending previous results with a new approach.

Findings

Jacobian convergence in L^1_{loc} under Sobolev convergence

Use of Federer's area formula to analyze measure convergence

Conditions ensuring measure-zero set preservation

Abstract

We prove that given a sequence of homeomorphisms $f_k: \Omega \to \mathbb{R}^n$ convergent in $W^{1,p}(\Omega, \mathbb{R}^n)$, $p \geq 1$ for $n =2$ and $p > n-1$ for $n \geq 3$, to a homeomorphism $f$ which maps sets of measure zero onto sets of measure zero, Jacobians $Jf_k$ converge to $Jf$ in $L^1_{loc}(\Omega)$. We prove it via Federer's area formula and investigation of when $|f_k(E)| \to |f(E)|$ as $k \to \infty$ for Borel subsets $E \Subset \Omega$.