Delaunay type domains for an overdetermined elliptic problem inn× ℝ and ℍn× ℝ

Abstract

We prove the existence of a countable family of Delaunay type domains \begin{eqnarray} \subset \mathbb{M}^n \times \mathbb{R}, \end{eqnarray} Ω t ⊂ M n × R , t ∈ℕ, where n is the Riemannian manifold n or ℍ n and n ≥ 2, bifurcating from the cylinder B n × ℝ (where B n is a geodesic ball in n ) for which the first eigenfunction of the Laplace–Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. In other words, the overdetermined problem \begin{eqnarray} \left\{ \begin{array} {ll} \Delta_g\, u + \gl\, u = 0 &\mbox{in }\; \Omega_t u=0 & \mbox{on }\; \partial g(\nabla u, \nu) = \const. &\mbox{on }\; \partial \end{array} \right. \end{eqnarray} See Formula in PDF has a bounded positive solution for some positive constant λ , where g is the standard metric in n × ℝ. The domains Ω t are rotationally symmetric and periodic with respect to the ℝ-axis of the cylinder and the sequence { Ω t } t converges to the cylinder B n × ℝ.