Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917902972616704 |
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| author | Dipasquale, Federico Luigi |
| author_facet | Dipasquale, Federico Luigi |
| contents | Relying on a recent criterion, due to A.~Petrunin [18], to check if a complete, non-compact, Riemannian manifold admits an isometric embedding into a Euclidean space with positive reach, we extend to manifolds with such property the singular extension results of B.~Bulanyi and J.~Van~Schaftingen [5] for maps in the critical, nonlinear Sobolev space $W^{m/(m+1),m+1}\left(X^m,\mathcal{N}\right)$, where $m \in \mathbb{N} \setminus \{0\}$, $\mathcal{N}$ is a compact Riemannian manifold, and $X^m$ is either the sphere $\mathbb{S}^m = \partial \mathbb{B}^{m+1}_+$, the plane $\mathbb{R}^m$, or again $\mathbb{S}^m$ but seen as the boundary sphere of the Poincaré ball model of the hyperbolic space $\mathbb{H}^{m+1}$. As in [5], we obtain that the extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate. Finally, we provide some illustrative examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05570 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds Dipasquale, Federico Luigi Analysis of PDEs Differential Geometry Functional Analysis 58D15 (Primary), 46E35 (Secondary) Relying on a recent criterion, due to A.~Petrunin [18], to check if a complete, non-compact, Riemannian manifold admits an isometric embedding into a Euclidean space with positive reach, we extend to manifolds with such property the singular extension results of B.~Bulanyi and J.~Van~Schaftingen [5] for maps in the critical, nonlinear Sobolev space $W^{m/(m+1),m+1}\left(X^m,\mathcal{N}\right)$, where $m \in \mathbb{N} \setminus \{0\}$, $\mathcal{N}$ is a compact Riemannian manifold, and $X^m$ is either the sphere $\mathbb{S}^m = \partial \mathbb{B}^{m+1}_+$, the plane $\mathbb{R}^m$, or again $\mathbb{S}^m$ but seen as the boundary sphere of the Poincaré ball model of the hyperbolic space $\mathbb{H}^{m+1}$. As in [5], we obtain that the extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate. Finally, we provide some illustrative examples. |
| title | Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry Functional Analysis 58D15 (Primary), 46E35 (Secondary) |
| url | https://arxiv.org/abs/2406.05570 |