Weakly stable solutions of Serrin's problem

Technical Summary: Weakly Stable Solutions of Serrin's Problem

Problem Statement
The paper addresses Serrin's overdetermined problem, which seeks a domain $\Omega \subset \mathbb{R}^n$ ($n \ge 2$) and a function $u \in C^\infty(\bar{\Omega})$ satisfying:
$$\begin{cases} -\Delta u = n & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \\ |Du| = 1 & \text{on } \partial\Omega. \end{cases}$$
While J. Serrin (1971) and H. Weinberger (1971) established that compact solutions must be unit balls, non-compact solutions exist (e.g., cylinders). H. Berestycki, L. Caffarelli, and L. Nirenberg conjectured that any non-compact solution must be either a unit ball or congruent to a cylinder of the form $\{y \in \mathbb{R}^m : |y| < m/n\} \times \mathbb{R}^{n-m}$. However, counterexamples to this conjecture exist for general semilinear problems. This paper investigates the conjecture specifically for the case $f(u) = n$ under the additional assumption of weak stability.

Methodology and Approach
The authors employ a variational approach combined with geometric analysis and asymptotic estimates. The core methodology involves:

1. Variational Characterization: Solutions are viewed as critical points of the functional $J(u) = \frac{1}{2}\int_{\{u>0\}} |Du|^2 - \int_{\{u>0\}} F(u)$ (where $F' = f = n$) subject to a relative volume constraint.
2. Stability Definitions:
   • Weak Stability: The domain satisfies the second derivative test for $J$ among compact perturbations preserving the relative volume of $\{u > 0\}$. This is formalized by the inequality:
     $$\int_{\partial\Omega} (f(0) - H) \phi^2 \le \int_{\Omega} (|D\phi|^2 - f'(u)\phi^2)$$
     for all $\phi \in C^\infty_c(\bar{\Omega})$ with $\int_{\partial\Omega} \phi = 0$.
   • Stability: The same inequality holds for all $\phi \in C^\infty_c(\bar{\Omega})$ without the zero-mean constraint.
3. Gradient and Asymptotic Estimates: The authors establish crucial gradient estimates for $u$ (specifically $|Du| < 1$ in $\Omega$ and $H \le n$ on $\partial\Omega$) and analyze the asymptotic behavior of the volume $V(\lambda)$ and boundary area $A(\lambda)$ of $\Omega$ intersected with large balls $B_\lambda(0)$.
4. Contradiction Argument: The proof proceeds by assuming $\Omega$ has infinite volume. Using the asymptotic relation $A(\lambda) \approx n V(\lambda)$ derived from the divergence theorem, the authors demonstrate that infinite volume implies the domain is fully stable. By testing the stability inequality with specific cutoff functions and utilizing the subharmonicity of an auxiliary function $\phi = u + \frac{1}{2}|Du|^2$, they derive a contradiction between the sign of the boundary integral and the stability condition.
5. Finite Volume Analysis: Once infinite volume is ruled out, the authors adapt Weinberger's argument to the non-compact setting to show that finite volume components must be unit balls. Weak stability is then used to prove the domain is connected, ruling out disjoint unions of balls.

Key Results

• Theorem 2 (Main Result): For any dimension $n \ge 2$, if $\Omega$ is a weakly stable solution of Serrin's problem, then $\Omega$ is a ball of radius 1. Consequently, no non-compact weakly stable solutions exist.
• Gradient Bounds: The paper proves that for any solution, $|Du| < 1$ in $\Omega$ and the mean curvature $H$ satisfies $H \le n$ on $\partial\Omega$.
• Stability Implication: If a solution has infinite volume, weak stability implies full stability.
• Dimensional Independence: Unlike previous results for related problems (e.g., the one-phase Bernoulli problem) which depend heavily on dimension (e.g., $n \le 3$ vs. $n \ge 7$), this result holds for all $n \ge 2$.

Significance and Claims
The paper claims to prove that weakly stable solutions of Serrin's problem are necessarily compact (specifically, round balls). This result effectively resolves the Berestycki-Caffarelli-Nirenberg conjecture for the case $f=n$ under the assumption of weak stability by demonstrating that the non-compact alternatives (cylinders) are not weakly stable. This assumption is described as "natural from a variational point of view."

The authors highlight that their result generalizes a previous theorem (Theorem 1 in [12]) which was restricted to dimension $n=2$ and required bounded curvature of $\partial\Omega$. The current work removes the dimension restriction and the curvature boundedness assumption. The proof technique is distinct from previous works; it avoids the Gauss-Bonnet theorem (specific to $n=2$) and arguments specific to low dimensions that fail in higher dimensions due to known counterexamples in related stability problems.

The result is situated within the context of minimal surface theory, where weakly stable solutions of Serrin's problem arise as blow-up limits of weakly stable minimal capillary surfaces. The rigidity result established here serves as a fundamental ingredient for characterizing such capillary surfaces with contact angles close to $0$ or $\pi$.