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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2402.17373 |
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| _version_ | 1866915797134213120 |
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| author | Detaille, Antoine |
| author_facet | Detaille, Antoine |
| contents | We consider the strong density problem in the Sobolev space $ W^{s,p}(Q^{m};\mathscr{N}) $ of maps with values into a compact Riemannian manifold $ \mathscr{N} $. It is known, from the seminal work of Bethuel, that such maps may always be strongly approximated by $ \mathscr{N} $-valued maps that are smooth outside of a finite union of $ (m -\lfloor sp \rfloor - 1) $-planes. Our main result establishes the strong density in $ W^{s,p}(Q^{m};\mathscr{N}) $ of an improved version of the class introduced by Bethuel, where the maps have a singular set without crossings. This answers a question raised by Brezis and Mironescu.
In the special case where $ \mathscr{N} $ has a sufficiently simple topology and for some values of $ s $ and $ p $, this result was known to follow from the method of projection, which takes its roots in the work of Federer and Fleming. As a first result, we implement this method in the full range of $ s $ and $ p $ in which it was expected to be applicable. In the case of a general target manifold, we devise a topological argument that allows to remove the self-intersections in the singular set of the maps obtained via Bethuel's technique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17373 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An improved dense class in Sobolev spaces to manifolds Detaille, Antoine Functional Analysis 46E35 We consider the strong density problem in the Sobolev space $ W^{s,p}(Q^{m};\mathscr{N}) $ of maps with values into a compact Riemannian manifold $ \mathscr{N} $. It is known, from the seminal work of Bethuel, that such maps may always be strongly approximated by $ \mathscr{N} $-valued maps that are smooth outside of a finite union of $ (m -\lfloor sp \rfloor - 1) $-planes. Our main result establishes the strong density in $ W^{s,p}(Q^{m};\mathscr{N}) $ of an improved version of the class introduced by Bethuel, where the maps have a singular set without crossings. This answers a question raised by Brezis and Mironescu. In the special case where $ \mathscr{N} $ has a sufficiently simple topology and for some values of $ s $ and $ p $, this result was known to follow from the method of projection, which takes its roots in the work of Federer and Fleming. As a first result, we implement this method in the full range of $ s $ and $ p $ in which it was expected to be applicable. In the case of a general target manifold, we devise a topological argument that allows to remove the self-intersections in the singular set of the maps obtained via Bethuel's technique. |
| title | An improved dense class in Sobolev spaces to manifolds |
| topic | Functional Analysis 46E35 |
| url | https://arxiv.org/abs/2402.17373 |