Generic topological screening and approximation of Sobolev maps
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| Format: | Preprint |
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2025
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| _version_ | 1866910805808644096 |
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| author | Bousquet, Pierre Ponce, Augusto C. Van Schaftingen, Jean |
| author_facet | Bousquet, Pierre Ponce, Augusto C. Van Schaftingen, Jean |
| contents | This manuscript develops a framework for the strong approximation of Sobolev maps with values in compact manifolds, emphasizing the interplay between local and global topological properties. Building on topological concepts adapted to VMO maps, such as homotopy and the degree of continuous maps, it introduces and analyzes extendability properties, focusing on the notions of $\ell$-extendability and its generalization, $(\ell, e)$-extendability.
We rely on Fuglede maps, providing a robust setting for handling compositions with Sobolev maps. Several constructions -- including opening, thickening, adaptive smoothing, and shrinking -- are carefully integrated into a unified approach that combines homotopical techniques with precise quantitative estimates.
Our main results establish that a Sobolev map $u \in W^{k, p}$ defined on a compact manifold of dimension $m > kp$ can be approximated by smooth maps if and only if $u$ is $(\lfloor kp \rfloor, e)$-extendable with $e = m$. When $e < m$, the approximation can still be carried out using maps that are smooth except on structured singular sets of rank $m - e - 1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generic topological screening and approximation of Sobolev maps Bousquet, Pierre Ponce, Augusto C. Van Schaftingen, Jean Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary: 58D15, 46E35, 58C25, Secondary: 47H11, 26A99, 55S35 This manuscript develops a framework for the strong approximation of Sobolev maps with values in compact manifolds, emphasizing the interplay between local and global topological properties. Building on topological concepts adapted to VMO maps, such as homotopy and the degree of continuous maps, it introduces and analyzes extendability properties, focusing on the notions of $\ell$-extendability and its generalization, $(\ell, e)$-extendability. We rely on Fuglede maps, providing a robust setting for handling compositions with Sobolev maps. Several constructions -- including opening, thickening, adaptive smoothing, and shrinking -- are carefully integrated into a unified approach that combines homotopical techniques with precise quantitative estimates. Our main results establish that a Sobolev map $u \in W^{k, p}$ defined on a compact manifold of dimension $m > kp$ can be approximated by smooth maps if and only if $u$ is $(\lfloor kp \rfloor, e)$-extendable with $e = m$. When $e < m$, the approximation can still be carried out using maps that are smooth except on structured singular sets of rank $m - e - 1$. |
| title | Generic topological screening and approximation of Sobolev maps |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary: 58D15, 46E35, 58C25, Secondary: 47H11, 26A99, 55S35 |
| url | https://arxiv.org/abs/2501.18149 |