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arXiv · 2501.11106

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

Zofia Grochulska
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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 $E \Subset \Omega$.

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Zofia Grochulska. 2025-01-19. $L^{1}_{loc}$-convergence of Jacobians of Sobolev homeomorphisms via area formula. https://arxiv.org/abs/2501.11106

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