Analytical obstructions to the weak approximation of Sobolev mappings into manifolds
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| Format: | Preprint |
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2024
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| _version_ | 1866909577721675776 |
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| author | Detaille, Antoine Van Schaftingen, Jean |
| author_facet | Detaille, Antoine Van Schaftingen, Jean |
| contents | For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ such that if $ \dim \mathcal{M} > p $, there is a map in the Sobolev space of mappings $ W^{1,p} (\mathcal{M}, \mathcal{N})$ which is not a weak limit of smooth maps into $ \mathcal{N} $ due to a mechanism of analytical obstruction. For $ p = 4n - 1 $, the target manifold can be taken to be the sphere $ \mathbb{S}^{2n} $ thanks to the construction by Whitehead product of maps with nontrivial Hopf invariant, generalizing the result by Bethuel for $ p = 4n -1 = 3$. The results extend to higher order Sobolev spaces $ W^{s,p} $, with $ s \in \mathbb{R} $, $s \geq 1 $, $ sp \in \mathbb{N}$, and $ sp \ge 2 $. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12889 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analytical obstructions to the weak approximation of Sobolev mappings into manifolds Detaille, Antoine Van Schaftingen, Jean Functional Analysis Analysis of PDEs 58D15 (Primary) 46E35, 46T10, 58C25 (Secondary) For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ such that if $ \dim \mathcal{M} > p $, there is a map in the Sobolev space of mappings $ W^{1,p} (\mathcal{M}, \mathcal{N})$ which is not a weak limit of smooth maps into $ \mathcal{N} $ due to a mechanism of analytical obstruction. For $ p = 4n - 1 $, the target manifold can be taken to be the sphere $ \mathbb{S}^{2n} $ thanks to the construction by Whitehead product of maps with nontrivial Hopf invariant, generalizing the result by Bethuel for $ p = 4n -1 = 3$. The results extend to higher order Sobolev spaces $ W^{s,p} $, with $ s \in \mathbb{R} $, $s \geq 1 $, $ sp \in \mathbb{N}$, and $ sp \ge 2 $. |
| title | Analytical obstructions to the weak approximation of Sobolev mappings into manifolds |
| topic | Functional Analysis Analysis of PDEs 58D15 (Primary) 46E35, 46T10, 58C25 (Secondary) |
| url | https://arxiv.org/abs/2412.12889 |