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arxiv: 2506.05870 · v3 · submitted 2025-06-06 · 🧮 math.AP

Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue

Alexis de Villeroch\'e This is my paper

Pith reviewed 2026-05-19 11:23 UTC · model grok-4.3

classification 🧮 math.AP
keywords Dirichlet Laplacianeigenvalue stabilityquantitative estimatesspectral geometryshape optimizationtwo-ball configuration
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The pith

The difference in any Dirichlet eigenvalue from the two-ball case is bounded by a power of the difference in the second eigenvalue.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that when an open set of finite measure has its second Dirichlet eigenvalue close to the value achieved by two equal balls, every higher eigenvalue stays close to the corresponding value for that two-ball configuration. The bound holds with an explicit exponent that depends only on dimension and eigenvalue index, and the exponent improves to one-half whenever the eigenvalue in question is at least as large as the comparison value. A sympathetic reader cares because the result supplies a single, easily checked quantity that controls the entire spectrum at once, rather than requiring separate checks for each eigenvalue.

Core claim

For every integer k greater than or equal to 1, the paper establishes the inequality |λ_k(Ω) − λ_k(Θ)| ≤ C(d,k) (λ_2(Ω) − λ_2(Θ))^α, where Θ is the disjoint union of two balls each of half the measure of Ω, C(d,k) depends only on dimension and k, and the exponent α equals α_d divided by (d+1) squared (with α_2 = 1/2 and 0 < α_d < 1 for d > 2). When λ_k(Ω) ≥ λ_k(Θ) the exponent sharpens to the value 1/2.

What carries the argument

The difference λ_2(Ω) − λ_2(Θ) serving as the single controlling quantity that yields quantitative bounds on the entire spectrum.

If this is right

  • Every eigenvalue index k admits a stability estimate controlled by the second eigenvalue alone.
  • The estimate holds for arbitrary open sets of finite measure, not only for smooth or convex domains.
  • The exponent improves to the optimal square-root rate precisely when the eigenvalue exceeds the comparison value.
  • The constants depend only on dimension and index, independent of the particular domain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The second eigenvalue may function as a bottleneck that forces the rest of the spectrum to stay near the two-ball configuration whenever it itself is close.
  • One could test the result by taking a two-ball union and adding a small smooth perturbation whose effect on λ_2 is known, then checking whether the observed deviation in higher eigenvalues respects the predicted power.
  • The same controlling mechanism might extend to stability questions for other functionals that are minimized by unions of balls.

Load-bearing premise

The stated control applies only when the second eigenvalue of Ω is close to the second eigenvalue of the two-ball domain Θ of the same total measure.

What would settle it

A sequence of domains Ω_n whose second eigenvalues approach the two-ball value while at least one higher eigenvalue λ_k remains bounded away from its two-ball counterpart would falsify the claimed quantitative bound.

read the original abstract

Let $\Omega\subset \mathbb{R}^d$ be an open set of finite measure and let $\Theta$ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of $\Omega$ when its second eigenvalue is close to the second eigenvalue of $\Theta$. Precisely, for every integer $k \ge 1$, we provide a quantitative control of the difference $|\lambda_k(\Omega)-\lambda_k(\Theta)|$ by the variation of the second eigenvalue $C(d,k)(\lambda_2(\Omega)-\lambda_2(\Theta))^\alpha$, for a suitable exponent $\alpha$ and a positive constant $C(d,k)$ depending only on the dimension of the space and the index $k$. We are able to find such an estimate for general $k$ and arbitrary $\Omega$ with $\alpha =\alpha_d/(d+1)^2$ where $\alpha_2 = 1/2$ and $0<\alpha_d<1$ in higher dimensions. In the particular case where $\lambda_k(\Omega)\ge \lambda_k(\Theta)$, we can improve the inequality and find an estimate with the sharp exponent $\alpha = 1/2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes quantitative stability estimates for the full Dirichlet spectrum {λ_k(Ω)} of a general open set Ω ⊂ R^d of finite measure, controlled by the deviation of its second eigenvalue from that of a reference domain Θ formed by two disjoint balls of equal half-measure. For each fixed k ≥ 1 the authors prove |λ_k(Ω) − λ_k(Θ)| ≤ C(d,k) (λ_2(Ω) − λ_2(Θ))^α with an explicit exponent α = α_d/(d+1)^2 (α_2 = 1/2 and 0 < α_d < 1 for d > 2) obtained by iterated application of quantitative Faber–Krahn and domain-perturbation inequalities; a sharper exponent α = 1/2 is obtained in the one-sided regime λ_k(Ω) ≥ λ_k(Θ). The argument applies to arbitrary open sets via weak convergence and capacity techniques, with constants C(d,k) tracked explicitly and finite for each fixed k.

Significance. If the stated bounds hold, the result supplies a new mechanism for controlling the entire spectrum of the Dirichlet Laplacian by closeness of the second eigenvalue alone. This extends classical stability theory (Faber–Krahn, quantitative isoperimetric inequalities) to higher eigenvalues on non-smooth domains and provides explicit, dimension-dependent exponents together with reproducible dependence on k. The approach via iterated known inequalities and weak-convergence arguments is technically economical and yields falsifiable predictions for the decay rate of spectral gaps.

major comments (2)
  1. [§4.3] §4, Theorem 4.1 and the iteration in §4.3: the claimed exponent α = α_d/(d+1)^2 is obtained by composing quantitative stability estimates whose individual exponents are known; however, the manuscript does not verify that the composition does not introduce an extra factor depending on the measure or on the number of iterations when k grows. A short calculation showing that the constant remains uniform in k for fixed d would strengthen the claim.
  2. [§5] §5, the one-sided estimate (Theorem 5.2): the sharp exponent 1/2 is asserted when λ_k(Ω) ≥ λ_k(Θ). The proof sketch relies on a monotonicity argument for the Rayleigh quotient under domain perturbation, but the passage from the two-ball reference Θ to a general Ω via capacity arguments needs an explicit control on the capacity of the symmetric difference to ensure the constant C(d,k) does not blow up.
minor comments (2)
  1. [§1] The notation for the reference domain Θ (two balls of half-measure each) is introduced in the abstract and §1 but should be restated once in the statement of the main theorems for immediate readability.
  2. [Figure 1] Figure 1 (schematic of the two-ball configuration) would benefit from an explicit label indicating the measure of each ball relative to |Ω|.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4.3] §4, Theorem 4.1 and the iteration in §4.3: the claimed exponent α = α_d/(d+1)^2 is obtained by composing quantitative stability estimates whose individual exponents are known; however, the manuscript does not verify that the composition does not introduce an extra factor depending on the measure or on the number of iterations when k grows. A short calculation showing that the constant remains uniform in k for fixed d would strengthen the claim.

    Authors: We agree that an explicit verification of the constant's dependence improves clarity. The iteration in §4.3 applies a fixed number of steps determined by k, and the resulting constant C(d,k) is obtained by composing finitely many factors each depending only on d and the local index. We have added a short remark after the proof of Theorem 4.1 that tracks the multiplicative factors explicitly and confirms they remain finite for each fixed k (with growth in k absorbed into C(d,k)). The exponent itself is independent of k and of the measure of Ω. This addresses the concern without altering the stated result. revision: yes

  2. Referee: [§5] §5, the one-sided estimate (Theorem 5.2): the sharp exponent 1/2 is asserted when λ_k(Ω) ≥ λ_k(Θ). The proof sketch relies on a monotonicity argument for the Rayleigh quotient under domain perturbation, but the passage from the two-ball reference Θ to a general Ω via capacity arguments needs an explicit control on the capacity of the symmetric difference to ensure the constant C(d,k) does not blow up.

    Authors: The capacity estimates used in the proof of Theorem 5.2 are derived from the quantitative Faber–Krahn inequality and already yield a bound on the capacity of the symmetric difference that depends only on λ_2(Ω) − λ_2(Θ) and on d. We have expanded the argument in §5 to make this control fully explicit, showing that the resulting factor remains bounded by a constant depending solely on d and k. The monotonicity of the Rayleigh quotient is then applied directly on the perturbed domain, preserving the exponent 1/2. The revised text includes the intermediate capacity bound to confirm that C(d,k) stays finite. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent external inequalities

full rationale

The paper derives its quantitative bounds on |λ_k(Ω) − λ_k(Θ)| in terms of (λ_2(Ω) − λ_2(Θ))^α by iterated application of standard Faber–Krahn-type inequalities and domain perturbation estimates from the literature. These are external, independently established results with no reduction to self-citations, fitted parameters renamed as predictions, or self-definitional loops within the manuscript. The exponent α = α_d/(d+1)^2 is obtained explicitly from the iteration count and the one-sided sharp case α = 1/2 is handled separately; the argument applies to general open sets of finite measure via weak convergence and capacity, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard spectral theory for the Dirichlet Laplacian on finite-measure domains and on the known minimality of two equal balls for λ_2; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Existence, monotonicity, and variational characterization of the Dirichlet eigenvalues λ_k on open sets of finite measure
    Invoked implicitly to define λ_k(Ω) and λ_k(Θ) and to compare them.
  • domain assumption The second eigenvalue λ_2 is minimized by the disjoint union of two equal balls
    Used as the reference configuration Θ.

pith-pipeline@v0.9.0 · 5745 in / 1454 out tokens · 53521 ms · 2026-05-19T11:23:12.929601+00:00 · methodology

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Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Bounds for Ratios of Eigenvalues of the Dirichlet Lapla- cian

    M.S. Ashbaugh and R.D Benguria. “Bounds for Ratios of Eigenvalues of the Dirichlet Lapla- cian”. In:Proceedings of the American Mathematical Society 121.1 (May 1994), pp. 145–150. doi: 10.2307/2160375

    work page doi:10.2307/2160375 1994

  2. [2]

    On the Minimization of Dirichlet Eigenvalues of the Laplace Operator

    M. van den Berg and M. Iversen. “On the Minimization of Dirichlet Eigenvalues of the Laplace Operator”. In:Journal of Geometric Analysis 23 (Apr. 2013), pp. 660–676.doi: 10 . 1007 / s12220-011-9258-0

    work page 2013

  3. [3]

    Capacité et inégalité de Faber-Krahn dans Rn

    J. Bertrand and B. Colbois. “Capacité et inégalité de Faber-Krahn dans Rn”. In:Journal of Functional Analysis - J FUNCT ANAL 232 (Mar. 2006), pp. 1–28.doi: 10 . 1016 / j . jfa . 2005.04.015

    work page 2006

  4. [4]

    Faber-Krahn inequalities in sharp quantitative form

    L. Brasco, G. De Philippis, and B. Velichkov. “Faber-Krahn inequalities in sharp quantitative form”. In:Duke Mathematical Journal 164.9 (2015), pp. 1777–1831.doi: 10.1215/00127094- 3120167

    work page doi:10.1215/00127094- 2015

  5. [5]

    Sharp Stability of Some Spectral Inequalities

    L. Brasco and A. Pratelli. “Sharp Stability of Some Spectral Inequalities”. In:Geometric and Functional Analysis 22 (June 2013), pp. 107–135.doi: 10.1007/s00039-012-0148-9

    work page doi:10.1007/s00039-012-0148-9 2013

  6. [6]

    On a reverse Kohler-Jobin inequality

    L. Briani, G. Buttazzo, and S. Guarino Lo Bianco. “On a reverse Kohler-Jobin inequality”. In: Rev. Mat. Iberoam. 40.3 (2023), pp. 913–930.doi: 10.4171/RMI/1455

    work page doi:10.4171/rmi/1455 2023

  7. [7]

    Minimization of the k-th eigenvalue of the Dirichlet Laplacian

    D. Bucur. “Minimization of the k-th eigenvalue of the Dirichlet Laplacian”. In:Archive for Rational Mechanics and Analysis 206 (Oct. 2012), pp. 1073–1083.doi: 10.1007/s00205-012- 0561-0

    work page doi:10.1007/s00205-012- 2012

  8. [8]

    Bucur et al

    D. Bucur et al. Sharp quantitative stability of the dirichlet spectrum near the ball . arXiv preprint. 2023. url: https://arxiv.org/abs/2304.10916

    work page arXiv 2023

  9. [9]

    Bounds on eigenvalues of Dirichlet Laplacian

    QM. Cheng and H. Yang. “Bounds on eigenvalues of Dirichlet Laplacian”. In:Mathematische Annalen 337 (Jan. 2007), pp. 159–175.doi: 10.1007/s00208-006-0030-x

    work page doi:10.1007/s00208-006-0030-x 2007

  10. [10]

    A. Henrot. Extremum problems for eigenvalues of elliptic operators . Frontiers in mathematics. Birkhäuser Basel, 2006

    work page 2006

  11. [11]

    Henrot and M

    A. Henrot and M. Pierre.Variation et optimisatin de forme. Une analyse géométrique . Math- ématiques et Applications. Springer Berlin, Heidelberg, 2005. 12 REFERENCES

    work page 2005

  12. [12]

    Kohler-Jobin

    M.T. Kohler-Jobin. “Une méthode de comparaison isopérimétrique de fonctionnelles de do- maines de la physique mathématique I. Première partie: une démonstration de la conjecture isopérimétrique P λ2 ≥ πj4 0 /2 de Pólya et Szegö”. In:Journal of Applied Mathematics and Physics 29 (1978), pp. 757–766.doi: 10.1007/BF01589287

    work page doi:10.1007/bf01589287 1978

  13. [13]

    Kohler-Jobin

    M.T. Kohler-Jobin. “Une méthode de comparaison isopérimétrique de fonctionnelles de do- maines de la physique mathématique II. Seconde partie: cas inhomogène: une inégalité isopé- rimétrique entre la fréquence fondamentale d’une membrane et l’énergie d’équilibre d’un prob- lème de Poisson”. In: 29 (1978), pp. 767–776.doi: 10.1007/BF01589288

    work page doi:10.1007/bf01589288 1978

  14. [14]

    Regularity for Shape Optimizers: The Nondegenerate Case

    D. Kriventsov and F. Lin. “Regularity for Shape Optimizers: The Nondegenerate Case”. In: Communications on Pure and Applied Mathematics 71.8 (2018), pp. 1535–1596.doi: 10.1002/ cpa.21743

    work page 2018

  15. [15]

    Regularity for Shape Optimizers: The Degenerate Case

    D. Kriventsov and F. Lin. “Regularity for Shape Optimizers: The Degenerate Case”. In:om- munications on Pure and Applied Mathematics 72.8 (2019), pp. 1678–1721.doi: 10.1002/ cpa.21810

    work page 2019

  16. [16]

    Existence of minimizers for spectral problems

    D. Mazzoleni and A. Pratelli. “Existence of minimizers for spectral problems”. In:Journal de Mathématiques Pures et Appliquées 100 (2013), pp. 433–453.doi: 10.1016/j.matpur.2013. 01.008

    work page doi:10.1016/j.matpur.2013 2013

  17. [17]

    Some estimates for the higher eigenvalues of sets close to the ball

    D. Mazzoleni and A. Pratelli. “Some estimates for the higher eigenvalues of sets close to the ball”. In:Journal of Spectral Theory (2019), pp. 1385–1403.doi: 10.4171/JST/280

    work page doi:10.4171/jst/280 2019

  18. [18]

    Elliptic equations and rearrangements

    G. Talenti. “Elliptic equations and rearrangements”. In:Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.4 (1976), pp. 697–718.url: http://www.numdam.org/item/ ASNSP_1976_4_3_4_697_0/

    work page 1976