Serrin's overdetermined problem on the sphere

TL;DR

This paper constructs new Serrin domains on the sphere that bifurcate from symmetric neighborhoods of the equator, providing the first examples not bounded by geodesic spheres, thus advancing understanding of overdetermined problems on curved surfaces.

Contribution

It introduces the first examples of Serrin domains on the sphere that are not bounded by geodesic spheres, using bifurcation from symmetric tubular neighborhoods.

Findings

Serrin domains exist on the sphere beyond geodesic sphere boundaries.

Bifurcation from symmetric neighborhoods leads to new solutions.

First known non-geodesic Serrin domains in spherical geometry.

Abstract

We study Serrin's overdetermined boundary value problem \begin{equation*} -\Delta_{S^N}\, u=1 \quad \text{ in $\Omega$},\qquad u=0, \; \partial_\eta u=\textrm{const} \quad \text{on $\partial \Omega$} \end{equation*} in subdomains $\Omega$ of the round unit sphere $S^N \subset \mathbb{R}^{N+1}$, where $\Delta_{S^N}$ denotes the Laplace-Beltrami operator on $S^N$. A subdomain $\Omega$ of $S^N$ is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in $S^{N}$ which are not bounded by geodesic spheres.