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Poincaré constants for solenoidal fields in n dimensions equal those for scalars in n+2 dimensions through a shared eigenfunction of the Laplace and Stokes operators.

2026-06-28 05:25 UTC pith:VOJBMTDR

load-bearing objection The paper computes explicit A- and n-dependent Poincaré constants for annuli and claims an exact match between nD Stokes constants and (n+2)D scalar Laplace constants via a shared eigenfunction relation.

arxiv 2606.04765 v1 pith:VOJBMTDR submitted 2026-06-03 math.AP

Exact Poincare Constants in n-dimensional Annuli

Bernd Rummler , Michael Ruzicka , Gudrun Th\"ater This is my paper
classification math.AP
keywords Poincaré constantsannuliStokes operatorLaplace operatoreigenvaluessolenoidal vector fieldsDirichlet boundary conditionsconcentric balls
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 calculates explicit Poincaré constants for annuli in dimensions n from 2 up to a finite N, where each annulus is the region between two concentric balls separated by a fixed gap of width 1. These constants depend explicitly on the nondimensional gap parameter A and on n. The central result is an exact match: the Poincaré constant for divergence-free vector fields with zero Dirichlet boundary values in R^n is identical to the constant for ordinary scalar functions with the same boundary condition in R^{n+2}. The match follows from the first eigenvalues of the scalar Laplacian and the Stokes operator being related by one common eigenfunction. The authors also treat the limiting regimes A approaching zero, where the annulus approaches a ball, and A approaching infinity, where a thin-gap approximation applies.

Core claim

We provide calculated (precise) Poincaré constants. These depend on A and the dimension n. Additionally we find a direct match of the Poincaré constants for solenoidal vector fields in R^n and the Poincaré constants for scalar functions in R^{n+2} (all with vanishing Dirichlet traces). This is based on the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator.

What carries the argument

The shared eigenfunction that equates the first eigenvalue of the scalar Laplace operator on an (n+2)-dimensional annulus with the first eigenvalue of the Stokes operator on the corresponding n-dimensional annulus.

Load-bearing premise

The lowest eigenvalue of the Stokes operator and the lowest eigenvalue of the scalar Laplacian are attained at the same function (up to the dimension shift).

What would settle it

Direct numerical computation of the smallest positive eigenvalue of the Stokes problem on a 3-dimensional annulus for a chosen A, compared against the smallest positive eigenvalue of the scalar Dirichlet Laplacian on the corresponding 5-dimensional annulus.

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

If this is right

  • The constants are available in closed form or by solving a one-dimensional ODE problem for every fixed n and A.
  • The eigenvalue on the annulus approaches the eigenvalue on the unit ball as A tends to zero, proved via the Green's function with Dirichlet conditions.
  • The small-gap limit A to infinity yields an asymptotic description of the constant.
  • Results for vector fields in any dimension can be read off from scalar results two dimensions higher, and vice versa.

Where Pith is reading between the lines

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

  • The dimension-shift identity may let known scalar eigenvalue bounds on balls be transferred directly to Stokes problems on lower-dimensional annuli.
  • The explicit A-dependence supplies a family of test domains for checking numerical eigenvalue solvers in both scalar and vector settings.

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 / 4 minor

Summary. The manuscript computes explicit (exact) Poincaré constants, i.e., reciprocals of the first Dirichlet eigenvalues, for the scalar Laplacian on n-dimensional annular domains Ω_{(n),A} (n=2,...,N) of fixed radial width 1 and inner radius A/2. It further asserts that these constants for solenoidal vector fields (Stokes operator) in dimension n coincide exactly with the scalar constants in dimension n+2, via a shared eigenfunction relating the first eigenvalues of the Laplace and Stokes operators. The paper also analyzes the limits A→0 (via Green's function on punctured domains Ω^*_{(n),σ} converging to the unit ball) and A→∞ (small-gap limit).

Significance. If the claimed explicit formulas and the dimensional correspondence hold, the results would supply closed-form Poincaré constants for a family of annular geometries that are otherwise only accessible numerically, together with a direct link between scalar and divergence-free problems that may simplify analysis in fluid mechanics and vector calculus on annuli. The limit statements recover known ball eigenvalues and small-gap asymptotics, providing consistency checks.

minor comments (4)
  1. §2 (or wherever the radial ODE is solved): the explicit formula for the first eigenfunction and eigenvalue on the annulus should be stated in closed form (Bessel or power functions) rather than left as the root of a transcendental equation; this would make the 'precise' claim verifiable without numerical root-finding.
  2. The proof of the n ↔ n+2 correspondence (abstract and §3) relies on a shared eigenfunction between the scalar Laplacian and the Stokes operator; the precise mapping of the vector field to the scalar function (including how the divergence-free condition is preserved) needs an explicit statement or reference to an equation.
  3. In the A→0 analysis, the convergence of the first eigenvalue on Ω^*_{(n),σ} to that of the unit ball is asserted via the Green's function; a quantitative rate or an explicit comparison inequality would strengthen the claim.
  4. Notation: the non-dimensional annulus definition (inner radius A/2, outer A/2+1) is clear, but the dependence of the constant on both A and n should be summarized in a single theorem statement for quick reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript on exact Poincaré constants for n-dimensional annuli and the dimensional correspondence between the Laplace and Stokes operators. We appreciate the recommendation of minor revision and note that no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper computes explicit Poincaré constants on annuli by solving the radial eigenvalue problems for the scalar Laplacian and the Stokes operator; the claimed dimensional match follows from an operator relation between the first eigenvalues and a shared eigenfunction, presented as a direct consequence rather than a fit or self-referential definition. No parameter is fitted to data and then relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The derivation chain remains self-contained against the stated boundary-value problems and the explicit radial symmetry of the domain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the eigenvalue correspondence between the Dirichlet Laplacian and the Stokes operator on annuli, plus the applicability of the Green's function representation in the A to 0 limit; no free parameters or new entities are mentioned.

axioms (1)
  • domain assumption The first eigenfunction of the scalar Laplacian on the annulus coincides with an eigenfunction of the Stokes operator, yielding identical Poincaré constants after dimension shift.
    Invoked to obtain the direct match between solenoidal vector fields in R^n and scalars in R^{n+2}.

pith-pipeline@v0.9.1-grok · 5806 in / 1397 out tokens · 32311 ms · 2026-06-28T05:25:51.481339+00:00 · methodology

0 comments
read the original abstract

We study $n$-dimensional annuli for $n\,\in\,\{2,\dots,N\}$ with $N\,<\,\infty$. We choose a non-dimensional setting such that for any fixed $n $ and given number ${\cal A}>0$ the annuli ${\Omega}_{(n),\cal A}$ are defined as space between two concentrical balls with radii ${\cal A}/2$ and ${\cal A}/2 +1$ in ${ R}^{n}$. For these geometries we provide calculated (precise) Poincar\'e constants. These depend on ${\cal A}$ and the dimension $n$. Additionally we find a direct match of the Poincar\'e constants for solenoidal vector fields in ${R}^{n}$ and the Poincar\'e constants for scalar functions in ${ R}^{n+2}$ (all with vanishing Dirichlet traces). This is based on the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator. In addition we consider the limit ${\cal A}\,\to\,0$. In this context problems in domains ${\Omega}_{(n),\sigma}^{*}$ are investigated. These domains enable us to use the Green's function of the Laplacian with vanishing Dirichlet traces to show that the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball in ${ R}^{n}$. On the other hand, we take advantage of the so-called small-gap limit for ${\cal A}\to\infty$.

Figures

Figures reproduced from arXiv: 2606.04765 by Bernd Rummler, Gudrun Th\"ater, Michael Ruzicka.

Figure 1
Figure 1. Figure 1: 2d Scetch of n-dimensional annulus with inner/outer radius To have a non-dimensional setting we use the following two well established frames: Either the so-called inverse relative gap 1Corresponding author E-mail: gudrun.thaeter@kit.edu 1 arXiv:2606.04765v1 [math.AP] 3 Jun 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Calculated (Laplace) Poincare constants for ´ n = 2, 3, 4, 5, 6, 7 as functions of A in logarithmic scale [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Calculated roots κ1,L(A) as functions of A in logarithmic scale n = 2, 3, 4, 5, 6, 7 References [1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equa￾tions satisfying general boundary conditions II, Comm. Pure Appl. Math. 17, 35-92 (1964). [2] T. Akinaga , S.C. Generalis , F.H. Busse; Tertiary and Quaternary States in the Taylor-Couette … view at source ↗
Figure 4
Figure 4. Figure 4: Zoomed in view of Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Zoomed in view of Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

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Forward citations

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