Serrin's result for hyperbolic space and sphere
The following result is proved: Theorem. Let \(\Omega\subset H^n\), where \(H^n\) is the \(n\)-dimensional hyperbolic space, be a bounded domain. Let \(u\in C^2(\overline\Omega)\) be a positive solution of \[ \Delta u+f(u)=0,\quad \text{in }\Omega, \] with boundary conditions \[ u=0,\quad {\partial u\over\partial\nu}=c\quad \text{on }\partial\Omega, \] where \(f\) is a \(C^1\) function, \(c\) is a constant and \(\partial u\over\partial\nu\) denotes the normal derivative along the inward normal field on \(\partial\Omega\). Then (i) \(\Omega\) is a geodesic ball and (ii) \(u\) is radially symmetric. This result generalizes a result of \textit{J. Serrin} [Arch. Ration. Mech. Anal. 43, 304-318 (1971; Zbl 0222.31007)] concerning a similar problem in \(\mathbb{R}^n\).
Laplace-Beltrami operator in hyperbolic spaces
scientific article; zbMATH DE number 1425071