Diffeomorphic approximation of piecewise affine homeomorphisms
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909828051369984 |
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| author | Campbell, Daniel D'Onofrio, Luigi Vítek, Tomáš |
| author_facet | Campbell, Daniel D'Onofrio, Luigi Vítek, Tomáš |
| contents | Given any $f$ a locally finitely piecewise affine homeomorphism of $Ω\subset \mathbb{R}^d$ onto $Δ\subset \mathbb{R}^d$ (for $d=3, 4$) such that $f\in W^{1,p}(Ω, \mathbb{R}^d)$ and $f^{-1}\in W^{1,q}(Δ, \mathbb{R}^d)$, $1\leq p ,q < \infty$ and any $ε>0$ we construct a diffeomorphism $\tilde{f}$ such that
$$\|f-\tilde{f}\|_{W^{1,p}(Ω,\mathbb{R}^d)} + \|f^{-1}-\tilde{f}^{-1}\|_{W^{1,q}(Δ,\mathbb{R}^d)} < ε.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diffeomorphic approximation of piecewise affine homeomorphisms Campbell, Daniel D'Onofrio, Luigi Vítek, Tomáš Analysis of PDEs Given any $f$ a locally finitely piecewise affine homeomorphism of $Ω\subset \mathbb{R}^d$ onto $Δ\subset \mathbb{R}^d$ (for $d=3, 4$) such that $f\in W^{1,p}(Ω, \mathbb{R}^d)$ and $f^{-1}\in W^{1,q}(Δ, \mathbb{R}^d)$, $1\leq p ,q < \infty$ and any $ε>0$ we construct a diffeomorphism $\tilde{f}$ such that $$\|f-\tilde{f}\|_{W^{1,p}(Ω,\mathbb{R}^d)} + \|f^{-1}-\tilde{f}^{-1}\|_{W^{1,q}(Δ,\mathbb{R}^d)} < ε.$$ |
| title | Diffeomorphic approximation of piecewise affine homeomorphisms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.02854 |