Current browse context:
math.AP
Change to browse by:
References & Citations
Mathematics > Analysis of PDEs
Title: Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space
(Submitted on 7 May 2024)
Abstract: We construct nontrivial unbounded domains $\Omega$ 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} -\Delta_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a positive bounded solution in $C^{2,\alpha}\left(\Omega\right) \cap H^1\left(\Omega\right)$. 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.
Submission history
From: Pieralberto Sicbaldi [view email][v1] Tue, 7 May 2024 14:26:05 GMT (40kb,D)
Link back to: arXiv, form interface, contact.