The extension of traces for Sobolev mappings between manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916329946087424 |
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| author | Van Schaftingen, Jean |
| author_facet | Van Schaftingen, Jean |
| contents | The compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ for which the trace operator from the first-order Sobolev space of mappings $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ to the fractional Sobolev-Slobodecki\uı space $\smash{\smash{\dot{W}}^{1 - 1/p, p}} (\partial \mathcal{M}, \mathcal{N})$ is surjective when $1 < p < \dim \mathcal{M}$ are characterised. The traces are extended using a new construction which can be carried out assuming the absence of the known topological and analytical obstructions. When $p \ge \dim \mathcal{M}$ the same construction provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18738 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The extension of traces for Sobolev mappings between manifolds Van Schaftingen, Jean Analysis of PDEs Functional Analysis 58D15 (Primary) 46E35, 46T10, 58C25, 58J32 (Secondary) The compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ for which the trace operator from the first-order Sobolev space of mappings $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ to the fractional Sobolev-Slobodecki\uı space $\smash{\smash{\dot{W}}^{1 - 1/p, p}} (\partial \mathcal{M}, \mathcal{N})$ is surjective when $1 < p < \dim \mathcal{M}$ are characterised. The traces are extended using a new construction which can be carried out assuming the absence of the known topological and analytical obstructions. When $p \ge \dim \mathcal{M}$ the same construction provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control. |
| title | The extension of traces for Sobolev mappings between manifolds |
| topic | Analysis of PDEs Functional Analysis 58D15 (Primary) 46E35, 46T10, 58C25, 58J32 (Secondary) |
| url | https://arxiv.org/abs/2403.18738 |