Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space
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| Format: | Preprint |
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2024
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| _version_ | 1866914787815849984 |
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| author | Dai, Guowei Sicbaldi, Pieralberto Zhang, Yong |
| author_facet | Dai, Guowei Sicbaldi, Pieralberto Zhang, Yong |
| contents | We construct nontrivial unbounded domains $Ω$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -Δ_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,Ω, \,\, u=0,\,\,\partial_νu=\text{const}\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a positive bounded solution in $C^{2,α}\left(Ω\right) \cap H^1\left(Ω\right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04348 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space Dai, Guowei Sicbaldi, Pieralberto Zhang, Yong Analysis of PDEs Differential Geometry We construct nontrivial unbounded domains $Ω$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -Δ_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,Ω, \,\, u=0,\,\,\partial_νu=\text{const}\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a positive bounded solution in $C^{2,α}\left(Ω\right) \cap H^1\left(Ω\right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space. |
| title | Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2405.04348 |