On the local character of the extension of traces for Sobolev mappings
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912159797084160 |
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| author | Van Schaftingen, Jean |
| author_facet | Van Schaftingen, Jean |
| contents | We prove that a mapping $u \colon \mathcal{M}'\to \mathcal{N}$, where $\mathcal{M}'$ and $ \mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U \colon \mathcal{M}' \times [0, 1) \to \mathcal{N}$ if and only if it is on some open covering of $\mathcal{M}'$. In the global case where $\mathcal{M}$ is a compact Riemannian manifold with boundary, this implies that the analytical obstructions to the extension of a mapping $u \colon \partial \mathcal{M}\to \mathcal{N}$ to some Sobolev mapping $U \colon \mathcal{M} \to \mathcal{N}$ are purely local. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12713 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the local character of the extension of traces for Sobolev mappings Van Schaftingen, Jean Analysis of PDEs Functional Analysis 58D15 (Primary) 46E35, 46T10, 58C25, 58J32 (Secondary) We prove that a mapping $u \colon \mathcal{M}'\to \mathcal{N}$, where $\mathcal{M}'$ and $ \mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U \colon \mathcal{M}' \times [0, 1) \to \mathcal{N}$ if and only if it is on some open covering of $\mathcal{M}'$. In the global case where $\mathcal{M}$ is a compact Riemannian manifold with boundary, this implies that the analytical obstructions to the extension of a mapping $u \colon \partial \mathcal{M}\to \mathcal{N}$ to some Sobolev mapping $U \colon \mathcal{M} \to \mathcal{N}$ are purely local. |
| title | On the local character of the extension of traces for Sobolev mappings |
| topic | Analysis of PDEs Functional Analysis 58D15 (Primary) 46E35, 46T10, 58C25, 58J32 (Secondary) |
| url | https://arxiv.org/abs/2412.12713 |