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

TL;DR

The paper addresses when a sequence of Sobolev homeomorphisms $f_k$ converging to $f$ in $W^{1,p}_{loc}$ yields convergence of image measures $|f_k(E)|\to|f(E)|$ and, crucially, $Jf_k\to Jf$ in $L^1_{loc}$. It develops a framework based on Federer's area formula, approximate differentiability, and the Lusin condition (N) to connect image-measure convergence to Jacobian convergence. Two main results establish that, for $n=2$ with $p\ge1$ and for $n\ge3$ with $p>n-1$, if $f$ satisfies (N) and the Sobolev convergence holds, then (eq1) holds for all Borel $E$, and the Jacobians converge in $L^1_{loc}$. The paper also provides a degree-theoretic criterion (Theorem T6) clarifying how non-injectivity interacts with area formula, and it proves a key folklore result that Sobolev homeomorphisms with the stated $p$-range converge locally uniformly, enabling the main convergence results. The combination of area formula, uniform convergence, and degree theory yields a robust understanding of Jacobian convergence under Sobolev limits, with implications for nonlinear elasticity and related fields.

Abstract

We prove that given a sequence of homeomorphisms $f_k: Ω\to \mathbb{R}^n$ convergent in $W^{1,p}(Ω, \mathbb{R}^n)$, $p \geq 1$ 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 $L^1_{loc}(Ω)$. 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 $E \Subset Ω$.

Theorems & Definitions (33)

• Example 1
• Theorem 2
• Corollary 3: HenclPratelli2018, MoraCorral2014, Theorem \ref{['1T2']}
• Theorem 4
• Example 5
• Theorem 6
• Theorem 8
• Theorem 9
• Definition 10
• Remark 11