$L^{1}_{loc}$-convergence of Jacobians of Sobolev homeomorphisms via area formula
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910973518938112 |
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| author | Grochulska, Zofia |
| author_facet | Grochulska, Zofia |
| contents | 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 $E \Subset Ω$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11106 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^{1}_{loc}$-convergence of Jacobians of Sobolev homeomorphisms via area formula Grochulska, Zofia Classical Analysis and ODEs 28A75 (Primary) 26B15 (Secondary) 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 $E \Subset Ω$. |
| title | $L^{1}_{loc}$-convergence of Jacobians of Sobolev homeomorphisms via area formula |
| topic | Classical Analysis and ODEs 28A75 (Primary) 26B15 (Secondary) |
| url | https://arxiv.org/abs/2501.11106 |