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arxiv: 1907.10957 · v1 · pith:6HMNOC43new · submitted 2019-07-25 · 🧮 math.AP

Sharp Estimates for the Principal Eigenvalue of the p-Operator

Thomas Koerber This is my paper

Pith reviewed 2026-05-24 16:13 UTC · model grok-4.3

classification 🧮 math.AP
keywords principal eigenvaluep-operatorcurvature-dimension conditionNeumann boundary conditionssharp estimatesintrinsic diameterp-Laplacian
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The pith

Under the BE(0,N) curvature condition the principal eigenvalue of the p-operator satisfies λ ≥ (p-1)π_p^p/D^p with equality only for BE(0,1).

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

The paper proves a sharp lower bound on the principal eigenvalue of a nonlinear p-operator that generalizes the p-Laplacian. It applies intrinsic Γ₂-calculus to an elliptic diffusion operator L on a compact manifold satisfying the curvature-dimension condition BE(0,N) and shows that the Neumann eigenvalue of the associated L_p is bounded below by (p-1) times π_p to the p divided by the square of the intrinsic diameter D. Equality holds exactly when the curvature condition reduces to the one-dimensional case BE(0,1). A separate lower bound is obtained for the real part of the principal eigenvalue of non-symmetric operators satisfying BE(a,∞).

Core claim

Given an elliptic diffusion operator L on a compact connected manifold with L-invariant measure m, the associated nonlinear p-operator L_p with Neumann boundary conditions has principal eigenvalue λ satisfying λ ≥ (p-1)π_p^p/D^p whenever L satisfies BE(0,N) for some N in [1,∞), with equality if and only if L satisfies BE(0,1). For a non-symmetric operator L = Δ_g + X·∇ satisfying BE(a,∞) the real part of its principal eigenvalue is at least π²/D² + a/2.

What carries the argument

The curvature-dimension condition BE(0,N) together with the intrinsic Γ₂-calculus applied to the nonlinear p-operator L_p.

If this is right

  • The bound is achieved precisely on one-dimensional spaces satisfying BE(0,1).
  • The same curvature assumption yields a lower bound π²/D² + a/2 on the real part of the principal eigenvalue for non-symmetric drift operators.
  • The estimates apply to manifolds that may have convex boundary in the sense compatible with the operator L.
  • The characterization of equality cases is complete under the stated curvature assumption.

Where Pith is reading between the lines

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

  • The result supplies a uniform way to obtain diameter-dependent eigenvalue controls once any manifold is shown to satisfy some BE(0,N).
  • The method may extend to other nonlinear generalizations of the Laplacian provided an analogous Γ₂-calculus identity can be derived.
  • On model spaces such as the circle or the interval the bound recovers the classical one-dimensional extremal value, confirming sharpness.

Load-bearing premise

The underlying diffusion operator L must satisfy the curvature-dimension condition BE(0,N) for some N on the compact manifold.

What would settle it

A compact manifold equipped with an operator L that violates BE(0,N) for every N yet has a Neumann eigenvalue of L_p strictly smaller than (p-1)π_p^p/D^p.

read the original abstract

Given an elliptic diffusion operator $L$ defined on a compact and connected manifold (possibly with a convex boundary in a suitable sense) with an $L$-invariant measure $m$, we introduce the non-linear $p-$operator $L_p$, generalizing the notion of the $p-$Laplacian. Using techniques of the intrinsic $\Gamma_2$-calculus, we prove the sharp estimate $\lambda\geq (p-1)\pi_p^p/D^p$ for the principal eigenvalue of $L_p$ with Neumann boundary conditions under the assumption that $L$ satisfies the curvature-dimension condition BE$(0,N)$ for some $N\in[1,\infty)$. Here, $D$ denotes the intrinsic diameter of $L$. Equality holds if and only if $L$ satisfies BE$(0,1)$. We also derive the lower bound $\pi^2/D^2+a/2$ for the real part of the principal eigenvalue of a non-symmetric operator $L=\Delta_g+X\cdot\nabla$ satisfying $\operatorname{BE}(a,\infty)$.

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

0 major / 3 minor

Summary. The manuscript introduces the nonlinear p-operator L_p associated to an elliptic diffusion operator L on a compact connected manifold (possibly with convex boundary) equipped with an L-invariant measure m. Under the curvature-dimension condition BE(0,N) for some N ∈ [1,∞), it establishes the sharp lower bound λ ≥ (p-1) π_p^p / D^p on the principal eigenvalue of L_p subject to Neumann boundary conditions, where D denotes the intrinsic diameter; equality holds if and only if L satisfies BE(0,1). A secondary bound π²/D² + a/2 is obtained for the real part of the principal eigenvalue of a non-symmetric operator L = Δ_g + X · ∇ satisfying BE(a,∞). The proofs rely on intrinsic Γ₂-calculus techniques.

Significance. If the derivations hold, the result supplies a parameter-free sharp eigenvalue bound that directly generalizes classical estimates for the p-Laplacian to a broad class of diffusion operators under explicit curvature-dimension hypotheses. The precise equality-case characterization (reduction to BE(0,1)) and the extension to non-symmetric operators constitute clear strengths. The bound is expressed solely in terms of the intrinsic diameter and the BE condition without auxiliary fitted quantities.

minor comments (3)
  1. [Abstract] The constant π_p appearing in the main estimate is not defined in the abstract or early introduction; an explicit one-line definition (the first positive zero of the one-dimensional p-sine function, or equivalent) should be supplied for readability.
  2. The precise definition of the p-operator L_p (including its action on test functions and the associated weak formulation) is referenced but not displayed in the provided abstract; placing the definition in §2 or as Eq. (1) would improve accessibility.
  3. [Abstract] The statement of the secondary result for non-symmetric operators does not specify the precise meaning of the principal eigenvalue (e.g., whether it is the eigenvalue of the linearized operator or a nonlinear analogue); a clarifying sentence would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on intrinsic Γ₂-calculus applied to the curvature-dimension condition BE(0,N) to obtain the eigenvalue lower bound λ ≥ (p-1)π_p^p/D^p, with equality characterized by reduction to BE(0,1). The bound is stated directly in terms of the given intrinsic diameter D and the external premise BE(0,N); no equation reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The additional bound for non-symmetric operators under BE(a,∞) follows the same pattern. The argument is self-contained against the stated assumptions and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the BE(0,N) curvature-dimension condition as the primary domain assumption and on the definition of the p-operator as a new construction. No free parameters are fitted to data; the bound is expressed in terms of p and the intrinsic diameter D. The p-operator is introduced to generalize the p-Laplacian but carries no independent evidence outside the paper.

axioms (2)
  • domain assumption The elliptic diffusion operator L satisfies the curvature-dimension condition BE(0,N) for some N in [1, ∞).
    Invoked explicitly as the hypothesis under which the eigenvalue estimate is proved.
  • domain assumption The manifold is compact and connected with an L-invariant measure m, possibly with convex boundary in a suitable sense.
    Standard setup stated for the domain of the operator L.
invented entities (1)
  • p-operator L_p no independent evidence
    purpose: Nonlinear generalization of the p-Laplacian to arbitrary elliptic diffusion operators L.
    Defined in the paper as the object whose principal eigenvalue is bounded; no falsifiable prediction outside the derivation is supplied.

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