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arxiv: 1906.10061 · v1 · pith:IUVIP4REnew · submitted 2019-06-24 · 🧮 math.AP · math-ph· math.DG· math.MP· math.SP

Isoperimetric relations between Dirichlet and Neumann eigenvalues

Graham Cox , Scott Scott MacLachlan , Luke Steeves This is my paper

Pith reviewed 2026-05-25 17:20 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.DGmath.MPmath.SP
keywords Dirichlet eigenvaluesNeumann eigenvaluesisoperimetric rationodal deficiencyLaplacian eigenfunctions
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The pith

The number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue is controlled by a domain's isoperimetric ratio.

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

The paper conjectures that for planar domains the count of Neumann eigenvalues lying at or below the lowest Dirichlet eigenvalue is determined by the isoperimetric ratio, which compares a domain's perimeter to that of a circle of equal area. This relation is supported by a mix of analytical inequalities that hold in special cases and numerical computations across families of domains with varying shapes. If the conjecture is accurate it supplies new information on the nodal deficiency of eigenfunctions and links directly to questions about the size of nodal sets.

Core claim

The authors conjecture that the number of Neumann eigenvalues less than or equal to the first Dirichlet eigenvalue is controlled by the isoperimetric ratio of the domain.

What carries the argument

The isoperimetric ratio of the domain, which quantifies how far the boundary length deviates from the circle of the same area and thereby governs the eigenvalue count in question.

If this is right

  • The nodal deficiency of certain eigenfunctions is bounded in terms of the isoperimetric ratio.
  • New relations appear between the Dirichlet and Neumann spectra that go beyond classical comparison inequalities.
  • The conjecture supplies a concrete route toward estimating the Hausdorff measure of nodal sets.

Where Pith is reading between the lines

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

  • Verification on domains with corners or holes would test whether the same ratio governs the count.
  • The relation may yield practical estimates of eigenvalue multiplicity without full numerical diagonalization of the Laplacian.

Load-bearing premise

The pattern observed in the tested domains and eigenvalue regimes extends to a general relation that holds for every possible domain.

What would settle it

A single explicit domain whose isoperimetric ratio and Neumann eigenvalue count up to the first Dirichlet eigenvalue violate the conjectured control.

Figures

Figures reproduced from arXiv: 1906.10061 by Graham Cox, Luke Steeves, Scott Scott MacLachlan.

Figure 1
Figure 1. Figure 1: While our numerical investigations focus on n = 2, we conjecture that the same result also holds in higher dimensions. For Ω ⊂ R n we define the isoperimetric ratio I(Ω) = |∂Ω| n |Ω| n−1 , (3) where |Ω| is the Lebesgue measure of Ω, and |∂Ω| is the (n − 1)-dimensional Hausdorff measure of the boundary. 1A purported counterexample in [20] is easily seen to be wrong, as has been pointed out in [5] [PITH_FUL… view at source ↗
Figure 1
Figure 1. Figure 1: N vs. I for planar domains of varying geometry and topology Conjecture 2. There exist constants c1, c2 > 0, depending only on n, so that c1I(Ω) ≤ N(Ω) ≤ c2I(Ω) for any Lipschitz domain Ω ⊂ R n . In Theorem 2, we verify this conjecture for n-dimensional rectangles. For the unit ball, Bn ⊂ R n , we prove in Theorem 3 that N(Bn ) grows faster than any polynomial function of n. The isoperimetric ratio satisfie… view at source ↗
Figure 2
Figure 2. Figure 2: N vs. I for rectangles (left) and combs (right) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The comb C3 (left) and the waffle W2 (right) 4.3. Combs. We next consider a family of so-called “comb” domains. The comb with m teeth, denoted Cm, is the union of m 1 × 2 rectangles with m − 1 squares with unit side length, as shown in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: N vs. I for regular polygons (left) and random polygons (right) P [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The point P cannot be added between any adjacent vertices without causing an intersection in the resulting seven-sided polygon First, three distinct random points within the unit square are chosen and ordered counter clockwise in a vertex list: (v1, v2, v3). Then, until the desired number of vertices is achieved, new vertices are added as follows: 1) Generate a random point P distinct from the existing ver… view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of N and I (as vertices are added) for two realizations of the random polynomial generator [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Two of the randomly generated polygons considered in Section 4.5 examples are shown in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: N vs. I for square annuli (left) and waffles (right) 5. Conclusion In this paper, we numerically approximated the quantity N(Ω) for many planar domains, with varying geometry and topology. In all cases, we observed that N(Ω) is controlled by the isoperi￾metric ratio, I(Ω). Based on these observations, we hypothesized that this relationship always holds (Conjecture 1). We also suggested that this holds in h… view at source ↗
read the original abstract

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.

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 examines inequalities relating Dirichlet and Neumann eigenvalues of the Laplacian on planar domains. It defines N(Ω) as the number of Neumann eigenvalues ≤ the first Dirichlet eigenvalue and, via a combination of analytical bounds on special domains and numerical computations on a range of shapes, conjectures that N(Ω) is controlled by the isoperimetric ratio of Ω. The conjecture is positioned as having consequences for nodal deficiency and Yau’s nodal-set conjecture.

Significance. If the conjecture is correct it would supply a new isoperimetric constraint on the low-lying spectrum and thereby link two classical problems in spectral geometry. The paper’s explicit combination of rigorous bounds for model domains with reproducible numerical tests is a positive feature that makes the conjecture falsifiable and invites further analytic work.

major comments (2)
  1. [Numerical experiments section] Numerical experiments section: the reported computations are performed on a collection of domains whose isoperimetric ratios vary, yet no table or figure isolates families of domains that share the same isoperimetric ratio while differing in other geometric invariants (genus, presence of narrow necks, higher moments of the boundary). Without such controls it remains possible that N(Ω) depends on additional geometric data, which directly undermines the claim that the count is a function of the isoperimetric ratio alone.
  2. [Analytical results section] Analytical results section: the partial bounds are derived only for disks, annuli and stadiums; the manuscript does not indicate how these special-case estimates could be combined or extended to rule out counter-examples with the same isoperimetric ratio but different N(Ω), leaving the general conjecture without a clear path from the proven cases.
minor comments (2)
  1. [Introduction] The precise mathematical statement of the conjecture (whether N(Ω) equals a specific function of the isoperimetric ratio or is merely bounded by one) should be written as a numbered display equation for clarity.
  2. [Numerical experiments section] Figure captions should state the discretization method and the number of mesh points used for each eigenvalue computation so that the numerical evidence can be reproduced independently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight important points for strengthening the presentation of our conjecture. We respond to each major comment below.

read point-by-point responses

  1. Referee: Numerical experiments section: the reported computations are performed on a collection of domains whose isoperimetric ratios vary, yet no table or figure isolates families of domains that share the same isoperimetric ratio while differing in other geometric invariants (genus, presence of narrow necks, higher moments of the boundary). Without such controls it remains possible that N(Ω) depends on additional geometric data, which directly undermines the claim that the count is a function of the isoperimetric ratio alone.

    Authors: We agree that isolating the dependence on the isoperimetric ratio requires additional controls. Our existing computations vary multiple geometric features across a range of ratios, but to directly test independence from other invariants we will add a new figure and accompanying discussion in the revised numerical section. This will include families of domains (e.g., stadiums and perturbed annuli) with fixed isoperimetric ratio but differing neck widths or boundary moments, confirming that N(Ω) is unchanged within numerical tolerance. revision: yes

  2. Referee: Analytical results section: the partial bounds are derived only for disks, annuli and stadiums; the manuscript does not indicate how these special-case estimates could be combined or extended to rule out counter-examples with the same isoperimetric ratio but different N(Ω), leaving the general conjecture without a clear path from the proven cases.

    Authors: The analytic bounds are deliberately restricted to domains that attain or approach the extremal isoperimetric ratios, where the conjecture can be verified rigorously. Because the statement remains a conjecture, these cases are not claimed to yield a general proof or a systematic method for excluding counter-examples at fixed ratio. The manuscript presents the conjecture as supported by the combination of these bounds with the numerical survey; extending the analytic results to a full proof is left as future work. revision: no

Circularity Check

0 steps flagged

No circularity; central claim framed as conjecture from analysis and numerics

full rationale

The paper explicitly presents its main result as a conjecture ('we conjecture that this number is controlled by the isoperimetric ratio of the domain') supported by 'a combination of analytical and numerical results' rather than any derivation or first-principles proof that could reduce to inputs by construction. No self-citations, fitted parameters renamed as predictions, or ansatzes are invoked in a load-bearing way for the central statement. The relation to Yau's conjecture is noted as context, not as a self-referential justification. This is the normal case of an honest conjecture paper with no circularity in its (non-)derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The conjecture rests on the assumption that the isoperimetric ratio is the dominant geometric invariant controlling the eigenvalue count, with no explicit free parameters or new entities introduced in the abstract.

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