Serrin's overdetermined problem on \(\mathbb{S}^N \times\mathbb{R}\) (Q6169168)

The author establishes the existence of non-trivial domains in \(\mathbb{S}^N \times\mathbb{R}\), \(N \geq 2\) which support a solution to the Serrin-type overdetermined boundary value problem \[ \begin{cases} \Delta_{\mathbb{S}^N \times\mathbb{R}} u=-1 & \text{ in }\Omega, \\ u=0 & \text{ on } \partial\Omega, \\ \frac{\partial u}{\partial \nu}=c & \text{ on } \partial \Omega. \end{cases} \] Here, \(\mathbb{S}^N\) is the \(N\)-dimensional unit sphere, \(c\) is a constant and \(\Delta_{\mathbb{S}^N \times\mathbb{R}}\) stands for the Laplace-Beltrami operator on \(\mathbb{S}^N \times\mathbb{R}\). Such domains bifurcate from symmetric straight tubular neighborhoods of \(\mathbb{S}^N \times \{ 0\}\), and are not bounded by geodesic spheres.