On the Schiffer and Berenstein conjectures with high-frequency for convex domains in the plane

Technical Summary: On the Schiffer and Berenstein Conjectures with High-Frequency for Convex Domains in the Plane

Problem Statement
This paper addresses two classical overdetermined elliptic problems in spectral geometry, known as the Schiffer and Berenstein conjectures. The central question is whether the existence of a nontrivial solution to these problems implies that the underlying bounded domain $\Omega \subset \mathbb{R}^2$ must be a disk.

The two problems are:

1. The Berenstein Conjecture (Dirichlet case): $-\Delta u = \alpha u$ in $\Omega$, with $u=0$ on $\partial\Omega$ and $\frac{\partial u}{\partial \nu} = c_D$ (constant) on $\partial\Omega$.
2. The Schiffer Conjecture (Neumann case): $-\Delta u = \alpha u$ in $\Omega$, with $\frac{\partial u}{\partial \nu} = 0$ on $\partial\Omega$ and $u = c_N$ (constant) on $\partial\Omega$.

While it is known that balls admit solutions, the conjecture posits that balls are the unique domains admitting such solutions. Previous results by Berenstein and Yang established that if a domain admits an infinite sequence of such eigenvalues tending to infinity, the domain must be a ball. However, the case of a single large eigenvalue remained open. Recent counterexamples to the unrestricted conjectures (in doubly connected or unbounded domains) highlight the need for controlled geometric classes and frequency thresholds.

Methodology
The authors' approach is rooted in the stationary phase method but introduces significant modifications to handle the case of a single large eigenvalue rather than a sequence.

1. Two-Point Stationary Phase Expansion:
   Unlike previous works that relied on the limit of an infinite sequence of eigenvalues (where parameters $t$ and $\lambda$ could vary independently to satisfy specific asymptotic conditions), this paper fixes the eigenvalue $\alpha$ as large but finite. They express $\alpha = \lambda^2 - t^2$ with $t \in \{-1, 1\}$ fixed and $\lambda = \sqrt{\alpha + 1}$ large.
   Using Green's identities, the existence of a solution is reduced to the vanishing of specific oscillatory integrals over the boundary $\partial\Omega$. The authors apply a uniform two-point stationary phase expansion to these integrals. For a uniformly convex domain, the phase function has exactly two non-degenerate critical points (antipodal points).

2. Two-Point Amplitude Defect ($B(\Omega)$):
   A key innovation is the introduction of the "two-point amplitude defect," defined as:
   $$B(\Omega) := \max_{\theta \in \mathbb{R}, t \in \{-1,1\}} \left| \sqrt{\rho(\theta)}e^{th'(\theta)} - \sqrt{\rho(\theta + \pi)}e^{-th'(\theta + \pi)} \right|$$
   where $h$ is the support function and $\rho = h + h''$ is the radius of curvature. The authors prove that $B(\Omega) = 0$ if and only if $\Omega$ is a disk.

3. Quantitative Geometric Estimates:
   The paper establishes a quantitative relationship between the defect $B(\Omega)$ and the geometric distance of $\Omega$ from a disk.

   • General Case: For domains in a controlled $C^m$ class ($m \ge 5$), they prove $B(\Omega) \ge c_m \min(d_{\text{disk}}(\Omega)^{p_m}, 1)$, where $p_m = \frac{m-1}{m-2}$ and $d_{\text{disk}}$ is the $C^2$ support-function distance. This relies on an even-odd decomposition of the support function and interpolation inequalities.
   • Centrally Symmetric Case: If $\Omega$ is centrally symmetric, the defect relates linearly to the Hausdorff distance from centered disks, yielding a stronger estimate.

4. High-Frequency Exclusion:
   By combining the stationary phase expansion (which shows the integral is dominated by the defect) with the geometric estimates, the authors derive a threshold. If the eigenvalue $\alpha$ exceeds a specific bound depending on the geometric class and the distance from a disk, the defect cannot be zero, and thus no solution exists.

Key Contributions and Results

• Theorem 1.1 (Berenstein Conjecture): For a bounded uniformly convex domain $\Omega \subset \mathbb{R}^2$ with $C^{2,\epsilon}$ boundary, if the Dirichlet problem (1.1) admits a nontrivial solution for a sufficiently large eigenvalue $\alpha > \alpha_0(\Omega)$, then $\Omega$ must be a disk.
• Theorem 1.2 (Schiffer Conjecture): Similarly, for a bounded uniformly convex domain with a Lipschitz connected boundary, if the Neumann problem (1.2) admits a nontrivial solution for a sufficiently large eigenvalue $\alpha > \alpha_0(\Omega)$, then $\Omega$ must be a disk.
• Regularity Dependence: The paper provides explicit cutoffs for the eigenvalue $\alpha$ based on the regularity $m$ of the domain. For a domain at distance $\epsilon$ from a disk in a $C^m$ class, the exclusion threshold scales as $O(1 + \epsilon^{-2p_m})$.
• Sharpness: The authors demonstrate via construction (Proposition 3.8) that the exponent $p_m = \frac{m-1}{m-2}$ is optimal for the geometric conversion from distance to defect under uniform $C^m$ control, even within the centrally symmetric subclass.

Significance and Claims
The paper claims a partial positive answer to the Schiffer and Berenstein conjectures in two dimensions. Its primary significance lies in:

1. Single Eigenvalue Rigidity: It removes the requirement for an infinite sequence of eigenvalues, proving that a single sufficiently large overdetermined eigenvalue is sufficient to force the domain to be a disk, provided the domain is uniformly convex and sufficiently smooth.
2. Quantitative Mechanism: It identifies a specific geometric mechanism—the two-point amplitude defect—that characterizes disks and quantifies how deviations from circularity prevent the existence of high-frequency solutions.
3. Resolution of High-Frequency Counterexamples: The results clarify why recent counterexamples (which exist for specific frequencies or unbounded domains) do not contradict the conjecture in the high-frequency regime for controlled convex classes. The paper establishes that while counterexamples may exist below a certain frequency threshold, they are excluded above a class-dependent cutoff.

The authors explicitly state that their results are restricted to the two-dimensional case and do not constitute an "all-frequency" rigidity theorem (i.e., they do not rule out the existence of a finite number of exceptional eigenvalues below the derived threshold). The threshold $\alpha_0(\Omega)$ depends on the specific geometry of the domain, not just the class, meaning the result is a "high-frequency exclusion" rather than a universal statement for all eigenvalues.