Extremal domains in $\mathbb{S}^2$: Geometric and Analytic methods

In this article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$\Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega,$$ subject to the boundary conditions $u=0$ on $\partial\Omega$ and $|\nabla u|$ being locally constant along $\partial\Omega$. We refer to such domains as $f$--extremal domains. In the first part of the paper, we extend the moving plane method in $\mathbb{S}^2$ and show that if an $f$--extremal domain $\Omega$ contains a simple curve of maximum points of $u$, then both $\Omega$ and $u$ are either rotationally symmetric or antipodally symmetric. Using the Alexandrov reflection method, we establish an analogous symmetry result for properly embedded constant mean curvature (CMC) surfaces with capillary boundaries that contain a simple curve of minimum distance to the origin (a neck). In the second part, we strengthen these conclusions for specific nonlinearities, including the eigenvalue problem ($f(x)=\lambda x$), the Serrin problem ($f(x)=\lambda x + c$), harmonic domains ($f(x)=c$), and nonlinearities of the form $f(x)=\lambda x + x^\beta$, for constants $\lambda \geq 2$, $c \geq 0$, and $\beta \in (0,1)$. In these cases, we prove that the domain $\Omega$ must be rotationally symmetric. Throughout the paper, we restrict our attention to the analytic setting in order to simplify the exposition and highlight the main ideas.