Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds
| Autor: | Dipasquale, Federico Luigi |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | Triggered by a recent criterion, due to A.~Petrunin [17], 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 Poincar\'{e} 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. Comment: Minor corrections |
| Databáze: | arXiv |
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