Serrin’s overdetermined problem on the sphere

Abstract

We study Serrin’s overdetermined boundary value problem $$\begin{aligned} -\Delta _{S^N}\, u=1 \quad \text { in }\Omega ,\quad u=0, \; \partial _\eta u={\text {const}} \quad \text {on }\partial \Omega \end{aligned}$$ 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.