Abstract
In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space \(\mathbb {H}^n\) supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show that symmetries of the boundary at infinity imply symmetries on the domain itself. In dimension two, we can strengthen our results proving that a connected domain \(\Omega \subset \mathbb {H}^2\) with \(C^2\) boundary whose complement is connected and supports a bounded positive solution u to an overdetermined problem, assuming natural conditions on the equation and the behaviour at infinity of the solution, must be either a geodesic ball or, a horodisk or, a half-space determined by a complete equidistant curve or, the complement of any of the above example. Moreover, in each case, the solution u is invariant by the isometries fixing \(\Omega \).
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Acknowledgements
The first author, José M. Espinar, is partially supported by Spanish MEC-FEDER Grant MTM2013-43970-P; CNPq-Brazil Grants 405732/2013-9 and 14/2012 - Universal, Grant 302669/2011-6 - Produtividade; FAPERJ Grant 25/2014 - Jovem Cientista de Nosso Estado. Alberto Farina is partially supported by the ERC grant EPSILON (Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities) and by the ERC grant COMPAT (Complex Patterns for Strongly Interacting Dynamical Systems). Laurent Mazet is partially supported by the ANR-11-IS01-0002 grant.
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Communicated by A. Neves.
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