Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space

TL;DR

This paper constructs the first known nontrivial solutions to overdetermined elliptic problems in hyperbolic spaces, revealing new geometric configurations and conditions for existence in various dimensions.

Contribution

It introduces a novel method to construct unbounded domains in hyperbolic space where overdetermined elliptic problems admit solutions, extending previous results to higher dimensions.

Findings

Existence of solutions in hyperbolic space for specific domains

Construction of domains bifurcating from the complement of a ball

Conditions under which solutions exist in higher dimensions

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.