A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domain

1. Problem Statement

The paper addresses the problem of establishing universal inequalities for the eigenvalues of the Laplacian operator under Neumann boundary conditions on bounded convex domains in Euclidean space $\mathbb{R}^n$.

• Context: For Dirichlet eigenvalues, universal inequalities (independent of the specific geometry of the domain) are well-established (e.g., by Payne, Pólya, and Weinberger).
• The Challenge: For Neumann eigenvalues, no such general inequalities exist for arbitrary domains. A result by Colin de Verdière shows that any finite sequence of non-negative numbers can be realized as the initial Neumann eigenvalues of some domain.
• The Gap: While geometric restrictions (specifically convexity) allow for some inequalities, existing results for Neumann eigenvalues on convex domains were either dimension-dependent with large constants or lacked a unified relationship between arbitrary indices $k$ and $l$. The paper aims to unify and generalize these inequalities.

2. Methodology

The author employs a combination of domain monotonicity, convex geometry, and spectral partitioning techniques. The proof strategy proceeds in three main stages:

A. Reduction to Orthotopes

Using Proposition 2.1 (based on Hatcher's work and the John Ellipsoid Theorem), the author demonstrates that any bounded convex domain $\Omega$ can be approximated by an orthotope (a generalized rectangle) $R$ such that:
$$\frac{1}{\sqrt{n}}R \subseteq \Omega \subseteq nR$$
This allows the problem to be reduced from general convex domains to the specific case of orthotopes, where eigenvalue calculations are more tractable.

B. Domain Monotonicity

The proof utilizes Theorem 2.2 (Funano, 2018), which states that for nested convex domains $\Omega \subseteq \Omega'$, the Neumann eigenvalues satisfy $\mu_k(\Omega') \leq c n^2 \mu_k(\Omega)$. This property allows the transfer of bounds between the original domain $\Omega$, the approximating orthotope $R$, and its scaled versions.

C. Convex Partitioning (The Core Innovation)

The central technical contribution involves constructing specific convex partitions of an orthotope. The author proves two key lemmas (3.1 and 3.2) via induction on dimension $n$:

1. Lemma 3.1: For any orthotope and indices $k \geq l$, there exists a convex partition into $l' \leq l$ subdomains such that the diameter of each subdomain is bounded by a function of $\mu_k$.
2. Lemma 3.2: Similarly, a partition exists into $k' \leq k$ subdomains with diameters bounded by a function of $\mu_l$.

Construction Technique:
The partition construction distinguishes two cases based on the smallest side length ($a_1$) of the orthotope relative to a radius $r$:

• Case 1 ($a_1$ is small): The problem is reduced to the $(n-1)$-dimensional face of the orthotope, utilizing the inductive hypothesis.
• Case 2 ($a_1$ is large): The author constructs a "core" subdomain $\tilde{\Omega}$ (points at distance $\geq r$ from the boundary) and places a maximal set of $2r$-separated points within it. Using Voronoi cells generated by these points, a convex partition is formed. The number of points is controlled using Kröger's inequality (Theorem 2.4), which bounds eigenvalues by volume.

D. Synthesis

The final bounds are derived by combining:

• Buser's Theorem (Theorem 2.5): Relates the $k$-th eigenvalue of a domain to the minimum first eigenvalue of its partition parts.
• Payne–Weinberger Inequality (Theorem 2.3): Provides a lower bound for the first eigenvalue of a convex domain based on its diameter ($\mu_1 \geq \pi^2 / \text{Diam}^2$).

3. Key Contributions and Results

The main result is Theorem 1.1, which establishes two-sided universal inequalities for Neumann eigenvalues $\mu_k(\Omega)$ on a bounded convex domain $\Omega \subset \mathbb{R}^n$.

The Theorem:
There exists a universal constant $c > 0$ such that for any $k \geq l$:
$$\frac{1}{c n^{10}} \left(\frac{k}{l}\right)^{\frac{2}{n}} \mu_l(\Omega) \leq \mu_k(\Omega) \leq c n^{10} \left(\frac{k}{l}\right)^{2} \mu_l(\Omega)$$

Specific Contributions:

1. Unification: The result unifies previous separate inequalities:
   • It generalizes the lower bound $\mu_k \geq c_1 k^{2/n} \mu_1$ (Funano, 2018).
   • It generalizes the upper bound $\mu_k \leq c_2 k^2 \mu_1$ (Liu, 2012).
   • It generalizes the ratio bound $\mu_{k+1} \leq c_3 n^4 \mu_k$ (Funano, 2018).
2. Dimensional Dependence: The inequalities include a dimensional constant $n^{10}$. While the author notes this is likely not optimal (previous results had $n^4$ or dimension-free constants for specific cases), the significance lies in the unified form relating arbitrary $k$ and $l$.
3. Extension to Higher Dimensions: While the author previously proved this for planar ($n=2$) domains, this paper provides the inductive construction necessary to generalize the result to arbitrary dimensions $n$.

4. Significance

• Theoretical Advancement: This work bridges a significant gap in spectral geometry by providing a comprehensive framework for Neumann eigenvalues on convex sets, a regime where universal inequalities are notoriously difficult to obtain due to the flexibility of the spectrum.
• Methodological Rigor: The inductive construction of convex partitions using Voronoi cells and maximal separated points offers a robust technique that could be applied to other spectral problems involving domain decomposition.
• Clarification of Geometry-Spectrum Relations: The paper reinforces the deep connection between the geometric properties of convex domains (specifically diameter and volume) and the distribution of their Laplacian eigenvalues, quantifying how the spectrum scales with the index $k$ and dimension $n$.

In summary, Funano establishes that despite the lack of universal inequalities for general domains, convexity imposes a strict structure on the distribution of Neumann eigenvalues, bounding the growth of $\mu_k$ relative to $\mu_l$ with explicit (though dimensionally heavy) constants.