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

1. Problem Statement

The paper investigates Overdetermined Elliptic Problems (OEPs) on the 2-sphere ($S^2$). Specifically, the authors study domains $\Omega \subset S^2$ that support positive solutions $u$ to the equation:
$$\begin{cases} \Delta u + f(u, |\nabla u|) = 0 & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \\ |\nabla u| = \text{const} & \text{on each connected component of } \partial\Omega. \end{cases}$$
Such domains are termed $f$-extremal domains. The central question is to determine the geometric structure (symmetry and topology) of $\Omega$ and the solution $u$ based on the properties of the nonlinearity $f$ and the topology of the set of maximum points of $u$, denoted as $\text{Max}(u)$.

The authors restrict their analysis to analytic solutions and finite domains (where the complement consists of a finite union of disjoint topological disks with analytic boundaries) to simplify technical regularity issues while focusing on the core geometric and analytic mechanisms.

2. Methodology

The paper employs a dual approach, combining Geometric Methods (specifically the Moving Plane Method) and Analytic Methods (comparison principles and gradient estimates).

A. Geometric Method: Extended Moving Plane Method

• Classical Approach: Traditionally, the Alexandrov reflection method (moving plane method) requires the domain's complement to contain a full geodesic or a half-geodesic to initiate the reflection.
• Extension: The authors extend this method to $S^2$ under weaker conditions. They prove that if the complement of the domain contains two antipodal points, the method can be initiated.
• Lifting Argument: To handle cases where the domain's geometry prevents direct reflection (e.g., when the "blocking" set is not a full geodesic), the authors lift the problem to the universal cover of $\mathbb{R}^3 \setminus L(x)$ (where $L(x)$ is the line through antipodal points). This allows the application of reflection arguments in a parameter space where the domain becomes a "strip" or "band," avoiding topological obstructions.
• Application to CMC Surfaces: This geometric framework is adapted to study Constant Mean Curvature (CMC) surfaces with capillary boundaries in $\mathbb{R}^3$, specifically those containing a "neck" (a curve of minimum distance to the origin).

B. Analytic Method: Pseudo-Radial Functions and Model Solutions

• Model Solutions: The authors construct a 2-parameter family of rotationally symmetric "model solutions" $(\Omega_{R,M,f}, u_{R,M,f})$ to the ODE associated with the PDE. These models serve as comparison targets.
• The $\tau$-Function: A key innovation is the introduction of the $\tau$-function, defined as a normalized boundary gradient:
  $$\tau(\Gamma) = \max_{p \in \Gamma} \frac{|\nabla u|^2(p)}{h_f(u_{\max})^2}$$
  where $h_f$ is a specific function derived from the model solutions on geodesic disks. This function establishes a canonical correspondence between general solutions and model solutions based on boundary data.
• Pseudo-Radial Functions: Using the Implicit Function Theorem, they define a mapping $\Psi$ (pseudo-radial function) that maps points in a general solution's domain to the height coordinate of a corresponding model solution.
• Gradient Estimates: By comparing the general solution $u$ with the model solution $\bar{u}$ via the function $F = |\nabla u|^2 - \bar{W}$ (where $\bar{W}$ is the model gradient squared), they derive a sharp gradient estimate. This relies on the sign of a specific curvature term $\mu$, which is controlled by assumptions on the nonlinearity $f$.

3. Key Contributions and Results

Theorem A: Structural Symmetry (Geometric)

• Hypothesis: Let $(\Omega, u)$ be an analytic solution where $\text{Max}(u)$ contains a simple closed curve $\gamma$ that is an isolated component of $\text{Max}(u)$.
• Result: The pair $(\Omega, u)$ is either antipodally symmetric or rotationally symmetric.
  • If rotationally symmetric, $\Omega$ is either a disk or an annulus.
• Significance: This result holds for any nonlinearity $f$ (no sign assumptions required) and relies purely on the topology of the maximum set. It generalizes previous results that required specific forms of $f$.

Corollary A: Positivity Condition

• If $f(x, 0) > 0$ for all $x > 0$, then $\text{Max}(u)$ cannot be a disk (due to the maximum principle). Thus, if $\text{Max}(u)$ contains a curve, the domain must be an annulus.
• This applies to classical cases: Eigenvalue problem ($f=\lambda x$), Serrin problem ($f=\lambda x + c$), Harmonic domains ($f=c$), and Allen-Cahn.

Theorem B: Geometric Rigidity for CMC Surfaces

• Context: Properly embedded CMC annuli in $\mathbb{R}^3 \setminus \{0\}$ with capillary boundaries.
• Hypothesis: The surface contains a "neck" (isolated component of minimum distance to the origin) satisfying specific separation conditions.
• Result: The surface is either antipodally symmetric or rotationally symmetric.
  • If rotational: It is part of a catenoid ($H=0$) or a Delaunay surface ($H \neq 0$).
  • If antipodally symmetric: The boundary spheres must have equal radii.

Theorem C: Strict Rotational Symmetry (Analytic)

• Hypothesis: $f$ satisfies specific growth conditions: $f(x) \geq x f'(x)$ and $f'(x) \geq 2$ for all $x > 0$ (or $f \equiv 1$).
• Result: If $\text{Max}(u)$ contains infinitely many points (implying a curve), the solution must be rotationally symmetric. Antipodal symmetry is ruled out.
• Significance: This strengthens Theorem A by eliminating the antipodal case for a broad class of nonlinearities using the analytic comparison method. It confirms that for these specific $f$, the only extremal domains with non-trivial maximum sets are rotationally symmetric annuli.

4. Significance and Impact

1. Unification of Methods: The paper successfully bridges the gap between geometric reflection techniques (Alexandrov) and analytic comparison techniques (P-functions, gradient estimates). It shows how geometric intuition can be rigorously formalized using analytic tools in the absence of global P-functions for general nonlinearities.
2. Generalization of Serrin's Theorem: It extends the classical Serrin rigidity result (which states that the only domain supporting a solution to the torsion problem is a ball) to the spherical setting and to domains with disconnected boundaries (annuli).
3. New Classification Tools: The introduction of the $\tau$-function and pseudo-radial functions provides a robust framework for classifying solutions to overdetermined problems on manifolds where explicit P-functions are unavailable.
4. CMC Surface Theory: The results provide new rigidity theorems for capillary CMC surfaces, linking the existence of "necks" directly to rotational or antipodal symmetry, extending previous work on minimal surfaces to the CMC case.
5. Topological Constraints: The paper clarifies the relationship between the topology of the set of maximum points ($\text{Max}(u)$) and the topology of the domain $\Omega$, proving that the presence of a "neck" or a curve of maxima forces the domain to be an annulus (under positivity conditions).

In summary, Espinar and Marín provide a comprehensive classification of $f$-extremal domains in $S^2$, demonstrating that the geometry of the solution's maximum set dictates the global symmetry of the domain, with the specific type of symmetry (rotational vs. antipodal) determined by the analytic properties of the nonlinearity $f$.