Delaunay type domains for an overdetermined elliptic problem in ${𝕊}^n\times ℝ$ and ${ℍ}^n\times ℝ$

Morabito, Filippo; Sicbaldi, Pieralberto

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 1, pp. 1-28.

We prove the existence of a countable family of Delaunay type domains

$$\begin{array}{c}\hfill {\Omega }_t\subset {𝕄}^n\times ℝ,\end{array}$$

$t\in \mathbb{N}$, where ${𝕄}^n$ is the Riemannian manifold ${𝕊}^n$ or ${ℍ}^n$ and $n\ge 2$, bifurcating from the cylinder $B^n\times ℝ$ (where $B^n$ is a geodesic ball in ${𝕄}^n$) 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

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\Delta }_{g}u+\lambda u=0\hfill & \text{in}{\Omega }_t\hfill \\ u=0\hfill & \text{on}\partial {\Omega }_t\hfill \\ g(\nabla u,\nu )=\text{const.}\hfill & \text{on}\partial {\Omega }_t\hfill \end{array}\right.\end{array}$$

has a bounded positive solution for some positive constant $\lambda$, where $g$ is the standard metric in ${𝕄}^n\times ℝ$. The domains $\Omega {}_t$ are rotationally symmetric and periodic with respect to the $ℝ$-axis of the cylinder and the sequence ${\left\{\Omega {}_t\right\}}_t$ converges to the cylinder $B^n\times ℝ$.

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