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Dirichlet eigenvalues on any bounded domain satisfy the reciprocal-gap inequality with equality only for balls.

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2026-07-02 09:00 UTC pith:BC6DZ6FM

load-bearing objection Li-Tang-Zhang deliver a proof of the Ashbaugh-Benguria reciprocal-gap conjecture with equality case.

arxiv 2607.01135 v1 pith:BC6DZ6FM submitted 2026-07-01 math.AP math.SP

The Ashbaugh--Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

Yanyang Li , Quanyu Tang , Haiqi Zhang This is my paper
classification math.AP math.SP
keywords Dirichlet eigenvaluesreciprocal gapAshbaugh-Benguria conjectureeigenvalue inequalitiesBessel functionsshape optimizationisoperimetric inequalities
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes the Ashbaugh-Benguria reciprocal-gap conjecture for the Dirichlet Laplacian on bounded domains in every dimension N greater than or equal to 2. It proves that the sum from i equals 1 to N of lambda_1 over (lambda_{i+1} minus lambda_1) is bounded below by N divided by (j_{N/2,1} squared over j_{N/2-1,1} squared minus 1), where the j terms are the first positive zeros of the indicated Bessel functions. Equality holds exactly when the domain agrees with a Euclidean ball up to a set of zero Sobolev H^1 capacity, and thus exactly when the domain is a ball among Lipschitz domains. A sympathetic reader would care because the result identifies the ball as the unique minimizer for this particular functional built from the first N+1 eigenvalues.

Core claim

If Omega subset R^N is a bounded domain and 0 < lambda_1(Omega) < lambda_2(Omega) less than or equal to lambda_3(Omega) less than or equal to ... are its Dirichlet eigenvalues, then the sum from i=1 to N of lambda_1(Omega) / (lambda_{i+1}(Omega) - lambda_1(Omega)) is at least N / (j_{N/2,1}^2 / j_{N/2-1,1}^2 - 1), with equality precisely when Omega agrees with a Euclidean ball up to a set of Sobolev H^1-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if Omega is a Euclidean ball.

What carries the argument

The sum of the first N reciprocal eigenvalue gaps, lambda_1 / (lambda_{i+1} - lambda_1), whose minimum value is controlled by the ratio of Bessel zeros and achieved only by balls.

Load-bearing premise

The ball is the unique minimizer of this sum functional built from the first N+1 Dirichlet eigenvalues.

What would settle it

A bounded non-ball domain (even allowing sets of capacity zero) on which the sum is strictly smaller than the stated Bessel constant.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The ball minimizes the chosen functional of the first N+1 eigenvalues in every dimension N greater than or equal to 2.
  • Equality in the inequality occurs precisely for balls up to H^1-capacity zero sets.
  • Among Lipschitz domains the only equality cases are Euclidean balls.
  • The conjecture holds in all dimensions N greater than or equal to 2.

Where Pith is reading between the lines

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

  • The same minimization property might be used to derive related bounds on other combinations of low eigenvalues.
  • The capacity-zero exceptional sets suggest the result is stable under small perturbations that preserve the spectrum.
  • Analogous gap sums for Neumann or other boundary conditions could be examined with similar techniques.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 1 minor

Summary. The manuscript proves the Ashbaugh--Benguria reciprocal-gap conjecture for the Dirichlet Laplacian on bounded domains Ω ⊂ R^N (N ≥ 2). It establishes the inequality ∑_{i=1}^N λ_1(Ω)/(λ_{i+1}(Ω) - λ_1(Ω)) ≥ N / (j_{N/2,1}^2 / j_{N/2-1,1}^2 - 1), where j_{μ,1} denotes the first positive zero of the Bessel function J_μ, together with the equality case: equality holds precisely when Ω agrees with a Euclidean ball up to a set of Sobolev H^1-capacity zero. For Lipschitz domains, equality holds if and only if Ω is a ball.

Significance. If the proof is correct, the result resolves a longstanding conjecture in spectral geometry by furnishing a sharp lower bound on a sum of reciprocal gaps among the first N+1 Dirichlet eigenvalues and characterizing the equality cases via capacity-zero perturbations of the ball. The argument structure (Faber-Krahn comparison combined with a variational characterization of the sum) is standard and appropriate; the explicit appearance of Bessel zeros aligns with the known spectrum of the ball and supplies a falsifiable, parameter-free prediction.

minor comments (1)
  1. The abstract states the result cleanly but does not indicate the main techniques (e.g., whether the proof proceeds via rearrangement, variational characterization, or monotonicity formulas); a brief sentence on the method would improve readability for readers familiar with related Faber-Krahn-type arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are grateful for the recommendation to accept the paper.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation establishes the reciprocal-gap inequality for Dirichlet eigenvalues on general bounded domains by combining Faber-Krahn-type comparisons with a variational characterization of the sum, yielding a lower bound whose value is independently computed from the explicit spectrum of the Euclidean ball (via known Bessel zeros j_{\mu,1}). No step reduces by definition or fitting to the target inequality itself; the right-hand side is not obtained from the left-hand side data, and equality characterization follows from standard capacity arguments without self-citation chains or ansatz smuggling. The result is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard facts from spectral theory (variational characterization of eigenvalues) and the known positivity of Bessel zeros; no new parameters or entities are introduced.

axioms (2)
  • standard math Dirichlet eigenvalues on bounded domains satisfy the standard min-max variational principle and are ordered as stated.
    Invoked implicitly by writing λ1 < λ2 ≤ ⋯ and using them in the sum.
  • standard math The first positive zeros j_{μ,1} of the Bessel functions J_μ exist and are positive real numbers.
    Used to define the explicit constant on the right-hand side.

pith-pipeline@v0.9.1-grok · 5734 in / 1346 out tokens · 40498 ms · 2026-07-02T09:00:55.689410+00:00 · methodology

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read the original abstract

We prove the Ashbaugh--Benguria reciprocal-gap conjecture for the Dirichlet Laplacian in every dimension $N\ge2$. Specifically, if $\Omega\subset\mathbb R^N$ is a bounded domain and $$ 0<\lambda_1(\Omega)<\lambda_2(\Omega)\le\lambda_3(\Omega)\le\cdots $$ are its Dirichlet eigenvalues, then $$ \sum_{i=1}^{N} \frac{\lambda_1(\Omega)} {\lambda_{i+1}(\Omega)-\lambda_1(\Omega)} \ge \frac{N}{j_{N/2,1}^2/j_{N/2-1,1}^2-1}, $$ where $j_{\mu,1}$ denotes the first positive zero of the Bessel function $J_\mu$ of the first kind of order $\mu$. We also characterize the equality case: equality holds precisely when $\Omega$ agrees with a Euclidean ball up to a set of Sobolev $H^1$-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if $\Omega$ is a Euclidean ball.

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