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

TL;DR

This paper investigates the symmetry properties of extremal domains on the sphere supporting solutions to overdetermined elliptic problems, extending classical methods and establishing new symmetry results for specific nonlinearities.

Contribution

It extends the moving plane method to $ ext{S}^2$, proves symmetry of extremal domains under certain conditions, and applies these results to various nonlinear elliptic problems.

Findings

Extends symmetry results to domains with simple maximum or minimum curves.

Shows extremal domains are rotationally symmetric for specific nonlinearities.

Establishes symmetry of constant mean curvature surfaces with capillary boundaries.

Abstract

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.