Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain

1. Problem Statement

The paper addresses the spectral geometry of the Laplacian operator on bounded convex domains in the Euclidean plane ($\mathbb{R}^2$). Specifically, it investigates the Neumann eigenvalue problem:
$$\begin{cases} -\Delta u = \mu u & \text{in } \Omega, \\ \partial_n u = 0 & \text{on } \partial\Omega, \end{cases}$$
where $\Omega$ is a bounded convex domain with a Lipschitz boundary, and $\partial_n$ is the outward normal derivative.

The eigenvalues are ordered as $0 = \mu_0(\Omega) < \mu_1(\Omega) \leq \mu_2(\Omega) \leq \dots \to \infty$. The central problem is to establish universal inequalities relating different eigenvalues $\mu_k(\Omega)$ and $\mu_l(\Omega)$ (where $k \geq l$) that depend only on the indices $k$ and $l$, and not on the specific geometry of the domain $\Omega$ or the dimension (in the planar case).

2. Methodology

The author employs a reduction strategy that transforms the problem from general convex domains to rectangular domains, leveraging several key tools from spectral geometry and convex analysis:

• Domain Approximation (Proposition 2.1): Utilizing Hatcher's result based on John's Theorem, any bounded convex domain $\Omega$ can be sandwiched between two rectangular domains (scaled by factors involving $\sqrt{n}$ and $n$).
• Domain Monotonicity (Theorem 2.2): A variant of domain monotonicity for Neumann eigenvalues states that if $\Omega \subseteq \Omega'$ are convex, then $\mu_k(\Omega') \lesssim \mu_k(\Omega)$ (up to a dimensional constant). This allows the author to transfer inequalities from rectangular domains to general convex domains.
• Partitioning Strategy: The core of the proof involves decomposing the domain $\Omega$ into a finite number of smaller convex subdomains $\{\Omega_i\}$.
  • Buser's Theorem (Theorem 2.6): Used to establish a lower bound for the $k$-th eigenvalue of the whole domain based on the first eigenvalues of the partitioned subdomains: $\mu_k(\Omega) \geq \min_i \mu_1(\Omega_i)$.
  • Payne–Weinberger Inequality (Theorem 2.3): Provides a lower bound for the first eigenvalue of a convex subdomain based on its diameter: $\mu_1(\Omega_i) \gtrsim 1/(\text{Diam}(\Omega_i))^2$.
  • Kröger's Bounds (Theorems 2.4 & 2.5): Used to relate eigenvalues to the diameter and volume of the domain, specifically to control the number of subdomains required in the partition.
• Geometric Construction: The proofs distinguish between two cases based on the aspect ratio of the rectangular domain:
  1. Thin domains ($a \leq 2R$): Partitioned by slicing the longer side into segments of length $\approx R$.
  2. Fat domains ($a > 2R$): Uses a maximal set of $2R$-separated points in a "core" subdomain to construct a Voronoi partition, ensuring the subdomains have controlled diameters.

3. Key Contributions and Results

The paper establishes two new universal inequalities for planar ($n=2$) bounded convex domains. The notation $\alpha \lesssim \beta$ implies $\alpha \leq c\beta$ for a universal constant $c$ independent of the domain and indices.

Theorem 1.1 (Upper Bound)

For any planar bounded convex domain $\Omega$ and integers $k \geq l$:
$$\mu_k(\Omega) \lesssim \left(\frac{k}{l}\right)^2 \mu_l(\Omega)$$

• Significance: This unifies and generalizes previous results. It recovers Liu's sharp dimension-free inequality ($\mu_k \lesssim k^2 \mu_1$) when $l=1$ and improves upon the author's previous work regarding the ratio of consecutive eigenvalues.

Theorem 1.2 (Reverse/Lower Bound)

For any planar bounded convex domain $\Omega$ and integers $k \geq l$:
$$\mu_k(\Omega) \gtrsim \frac{k}{l} \mu_l(\Omega)$$

• Significance: This provides a sharp lower bound generalizing the author's previous result ($\mu_k \gtrsim k^2 \mu_1$ in $\mathbb{R}^n$) specifically for the planar case. It establishes that the growth of Neumann eigenvalues in planar convex domains is at least linear with respect to the index ratio.

Theorem 4.1 (Weaker Variant of Known Inequality)

The paper provides a new, shorter proof for a weaker version of a known inequality for general dimensions ($n$):
$$\mu_{k+1}(\Omega) \lesssim n^7 \mu_k(\Omega)$$

• Methodology: This proof utilizes the relationship between Neumann eigenvalues on a rectangle and the eigenvalues of the Laplacian on a flat torus, applying P. Li's theorem on homogeneous manifolds to show the ratio is bounded.

4. Significance and Impact

• Unification: The results unify disparate inequalities in spectral geometry, providing a coherent framework for understanding the distribution of Neumann eigenvalues on convex sets in 2D.
• Sharpness: The inequalities are described as "sharp" in the context of dimension-free bounds, offering precise control over the spectral gap and growth rates.
• Methodological Innovation: The use of a specific partitioning argument (combining Voronoi cells and diameter control) to reduce general convex domains to rectangular ones offers a robust technique that could be applicable to other spectral problems.
• Simplification: The paper offers a significantly shorter proof for a known inequality (Theorem 4.1), demonstrating the efficiency of the proposed geometric reduction method.

In summary, Kei Funano's paper advances the understanding of the spectral properties of the Laplacian on convex planar domains by proving that the ratio of any two Neumann eigenvalues is bounded by a function of their indices alone, specifically quadratic for the upper bound and linear for the lower bound.