Serrin’s Overdetermined Problem on ${{\mathbb {S}}}^N \times {{\mathbb {R}}}$

Abstract

We show the existence of non-trivial domains $\Omega$ of $\mathbb S^N \times {{\mathbb {R}}}$ (${\mathbb {S}}^N$ being the N-dimensional unit sphere) which support the solution to the Serrin’s overdetermined boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _{{\mathbb {S}}^N \times {{\mathbb {R}}}}u=-1 &{} \text{ in }\, \Omega ,\\ u=0 &{} \text{ on }\, \partial \Omega ,\\ \frac{\partial u}{\partial \nu }=const.&{} \text{ on }\, \partial \Omega .\\ \end{array} \right. \end{aligned}$$

Here $\Delta _{{\mathbb {S}}^N \times {{\mathbb {R}}}}$ denotes the Laplace–Beltrami operator on ${\mathbb {S}}^N \times {{\mathbb {R}}}$ and $\frac{\partial }{\partial \nu }$ denotes the derivative in the direction of the outer unit normal vector to $\partial \Omega .$ These domains are obtained by bifurcation of symmetric straight tubular neighborhoods of $\mathbb S^N \times \{0\}$ and they are not bounded by geodesic spheres.