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Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

This paper refines Kröger's sharp upper bounds for the Neumann eigenvalues of convex domains by establishing quantitative inequalities that incorporate the domain's diameter and the second largest semiaxis of its John ellipsoid, with an explicit constant provided for the planar case when $k=1$ .

Original authors: Dorin Bucur, Andrea Gentile, Antoine Henrot

Published 2026-04-16

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CC BY 4.0

Original authors: Dorin Bucur, Andrea Gentile, Antoine Henrot

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a drum. When you hit it, it vibrates and produces a sound. The pitch of that sound depends on the shape of the drum. In mathematics, these pitches are called eigenvalues. The lowest pitch is the "fundamental tone," and the higher ones are overtones.

This paper is about a specific type of drum: one with a convex shape (like a circle, a square, or an egg—no dents or holes) and a special rule: the edge of the drum is perfectly smooth, so the vibration doesn't "leak" out. This is called a Neumann boundary condition.

For a long time, mathematicians knew the "worst-case scenario" for how high the pitch could go. If you take a drum and squash it into a very long, thin line (like a piece of string), the pitch gets higher and higher, approaching a theoretical maximum limit. This limit was calculated by a mathematician named Kröger in 1999.

The Problem:
The paper asks: What happens if our drum is almost, but not quite, that thin string?
If the pitch is very close to the maximum limit, how close is the shape to being a flat line?

The Analogy: The "Flatness" Penalty
Think of the drum's shape as a piece of clay.

The Limit: If you roll the clay into a long, thin snake, the pitch hits the ceiling (the maximum).
The Reality: You can't actually make a perfect snake with a real drum (it needs some width).
The Discovery: The authors prove that if the pitch is slightly lower than the ceiling, it's not just because of random luck. It's because the drum has a tiny bit of "width" or "thickness."

They found a precise formula that acts like a penalty system.
$\text{Max Pitch} - \text{Actual Pitch} \approx \text{Constant} \times (\text{Thickness})^2$

In plain English: The amount by which your drum's pitch falls short of the maximum is directly related to the square of its thickness.

If you double the thickness of your drum, the pitch drops by four times as much.
If you make it ten times thicker, the pitch drops by a hundred times.

The "John Ellipsoid" Metaphor
To measure this "thickness," the authors use a clever geometric tool called the John Ellipsoid. Imagine trying to fit the biggest possible football (ellipsoid) inside your drum shape.

The longest axis of this football is the drum's length (diameter).
The second longest axis is the drum's "width" or "flatness."

The paper says: The more "flat" your drum is (the smaller that second axis is), the closer your pitch gets to the theoretical maximum.

Why is this important?
Usually, in math, we say "If X is true, then Y is true." But this paper goes further. It says, "If X is almost true, then Y is almost true, and here is exactly how much it differs based on how 'almost' you are."

This is a Quantitative Inequality. It's not just a rule; it's a ruler. It allows scientists to estimate how "thin" a shape is just by listening to its vibration, or conversely, predict how much the vibration will drop if they know the shape's thickness.

The "Planar" Bonus
For the special case of 2D shapes (like a flat drum on a table), the authors didn't just prove the rule exists; they actually calculated the exact number for the "penalty constant." They found that for a specific type of symmetric drum, the pitch drops by about 0.432 times the square of the width.

Summary
Think of this paper as a new law of physics for vibrating shapes:

The Limit: There is a maximum pitch a convex shape can reach, achieved only by an infinitely thin line.
The Cost: Any real shape has thickness.
The Rule: The "cost" of having thickness is that your pitch drops. The drop is proportional to the square of the thickness.
The Result: We now have a precise mathematical way to measure how "thin" a shape is just by looking at its vibration, or how much its vibration will suffer if we make it slightly thicker.

It's like realizing that if you want to run a race at the world record speed, you can't be wearing heavy boots. If you wear boots that are just 1mm thick, you will be slower by a specific, calculable amount. This paper tells us exactly what that amount is for the "race" of vibrating shapes.

1. Problem Statement

The paper addresses the optimization of the $k$ -th Neumann eigenvalue, denoted $\mu_k(\Omega)$ , for a bounded, open, and convex domain $\Omega \subset \mathbb{R}^d$ . Specifically, it focuses on refining the sharp upper bounds established by Kröger (1999).

Kröger proved that for any convex domain $\Omega$ with diameter $D_\Omega$ :
$D_\Omega^2 \mu_k(\Omega) < \mu^*_{k,d}$
where $\mu^*_{k,d}$ is a sharp constant depending only on the dimension $d$ and the index $k$ . The equality is never attained for $d \ge 2$ ; rather, the bound is approached asymptotically by a sequence of "collapsing" domains that converge to a line segment.

The Core Question: The authors seek a quantitative version of this inequality. They aim to establish an inequality of the form:
$D_\Omega^2 \mu_k(\Omega) \le \mu^*_{k,d} - C(k, d) \frac{a_2(\Omega)^2}{D_\Omega^2}$
where $a_2(\Omega)$ represents the "flatness" or the second largest semi-axis of the John ellipsoid of $\Omega$ . The goal is to quantify how much the eigenvalue drops below the sharp bound as the domain deviates from being a degenerate segment (i.e., as it becomes "thicker").

2. Methodology

The proof strategy relies on a combination of contradiction arguments, asymptotic analysis of collapsing domains, and variational methods.

A. Asymptotic Analysis of Collapsing Domains

The authors utilize the theory of collapsing convex sets. If a sequence of convex sets $\{\Omega_n\}$ with fixed diameter collapses to a lower-dimensional set, the eigenvalues converge to those of a weighted Sturm-Liouville problem on the limit set.

Limit Geometry: The limit of a collapsing sequence is a segment (for $k=1$ ) or a specific 1D structure.
Weighted Eigenvalues: The limit problem involves a weight function $p(x)$ derived from the cross-sectional area of the collapsing domain. The authors use results from [2] and [17] to characterize the eigenvalues of these weighted problems.

B. Proof by Contradiction (Theorem 1.1)

To prove the quantitative inequality, the authors assume the contrary: that there exists a sequence of domains where the eigenvalue is arbitrarily close to the upper bound relative to the flatness term.

Assumption: Assume $\mu_k(\Omega_n) \ge \mu^*_{k,d} - \frac{1}{n} a_2(\Omega_n)^2$ .
Convergence: By compactness properties of convex sets (Blaschke selection theorem), the sequence converges to a limit shape (a segment).
Test Function Construction: The core of the proof involves constructing a specific test function $w_n$ for the Rayleigh quotient. This function is a perturbation of the first eigenfunction of the limit Sturm-Liouville problem:
$w_n(x_1, x') = u_n(x_1) \left[ 1 + \tau (x_2 - c(x_1))^2 \right] - d_n$
where $u_n$ is the 1D eigenfunction, $c(x_1)$ is the centroid of the cross-section, and $\tau$ is a carefully chosen negative constant.
Expansion: The authors perform a second-order expansion of the Rayleigh quotient in terms of the small parameter $a_2(\Omega_n)$ .
Contradiction: They show that for a sufficiently small negative $\tau$ , the perturbed Rayleigh quotient yields a value strictly lower than the assumed bound, contradicting the minimality of $\mu_k$ . This relies on proving that a specific integral term (denoted $\gamma_n$ ) is strictly non-zero.

C. Explicit Constant Calculation (Theorem 1.3)

For the planar case ( $d=2$ ) and $k=1$ , the authors move beyond the abstract contradiction proof to derive an explicit constant.

Symmetry Assumption: They restrict the domain to be symmetric with respect to the mediatrix of its diameter.
Variational Problem: They formulate a calculus of variations problem to minimize a functional involving Bessel functions ( $J_0$ and $J_1$ ).
Optimization: By analyzing the derivative of the functional and using properties of Bessel functions, they prove that the optimal profile function is linear ( $h(x) = 2x$ ), corresponding to a triangle. This allows them to compute a concrete numerical value for the constant.

3. Key Contributions and Results

Theorem 1.1 (General Quantitative Inequality)

For any $k \in \mathbb{N}$ and $d \ge 2$ , there exists a constant $C(k, d) > 0$ such that for any convex domain $\Omega$ :
$D_\Omega^2 \mu_k(\Omega) \le \mu^*_{k,d} - C(k, d) \frac{a_2(\Omega)^2}{D_\Omega^2}$
This establishes that the "distance" from the optimal bound is controlled by the square of the second semi-axis of the John ellipsoid.

Theorem 1.2 (Sharpness of the Exponent)

The exponent 2 on the flatness term $a_2(\Omega)$ is sharp.

Proof: The authors construct a sequence of isosceles triangles (superequilateral) where the width $w_\Omega$ tends to zero. They show that the eigenvalue behaves as $\mu_1 \approx \mu^* - C w_\Omega^2$ , proving that a higher power (e.g., $w^3$ ) would not hold, and a lower power would be too weak.

Theorem 1.3 (Explicit Constant in 2D)

For a planar convex domain $\Omega$ symmetric with respect to the mediatrix of its diameter:
$D_\Omega^2 \mu_1(\Omega) \le 4j_{0,1}^2 - 0.432 \frac{w_\Omega^2}{D_\Omega^2}$
where $j_{0,1}$ is the first zero of the Bessel function $J_0$ , and $w_\Omega$ is the width orthogonal to the diameter.

Significance: While the constant $0.432$ is not proven to be the absolute sharp constant for all convex domains, it is an explicit, computable value derived from the worst-case geometry (triangles). This contrasts with previous results where constants were often extremely small or non-explicit (e.g., $10^{-16}$ ).

4. Significance and Impact

Bridging Geometry and Spectral Theory: The paper provides a rigorous link between the spectral gap of the Neumann Laplacian and the geometric "flatness" of the domain. It quantifies the intuition that "thin" domains have eigenvalues close to the Kröger bound.
Novelty in Quantitative Estimates: Most quantitative isoperimetric inequalities (like the quantitative Faber-Krahn) measure distance from a ball using the Fraenkel asymmetry. This paper is significant because the "target" is a degenerate segment (which has no interior), and the distance is measured by the second semi-axis. Such results for non-compact target sets are rare in the literature.
Explicit Constants: By providing an explicit constant ($0.432$) for the planar case, the authors move the field from purely existential proofs (which guarantee a constant exists but don't say what it is) to practical, usable estimates.
Methodological Advancement: The technique of using specific perturbations of eigenfunctions in collapsing domains to derive quantitative stability estimates offers a template for analyzing other spectral problems where the optimizer is a degenerate shape.

In summary, this work refines the classical Kröger inequality by adding a stability term that quantifies the deviation from the optimal (degenerate) geometry, proving the optimality of the quadratic decay rate, and providing explicit constants for the planar case.