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2025-05-30 09:59:45

This https://arxiv.org/abs/2501.11106 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

$L^{1}_{loc}$-convergence of Jacobians of Sobolev homeomorphisms via area formula
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 $E \Subset Ω$.