Maximization of the efficiency of the first Dirichlet eigenfunction and improved eigenvalue inequalities

Francesco Della Pietra

arxiv: 2604.22410 · v1 · submitted 2026-04-24 · 🧮 math.AP · math.OC

Maximization of the efficiency of the first Dirichlet eigenfunction and improved eigenvalue inequalities

Francesco Della Pietra This is my paper

Pith reviewed 2026-05-08 10:34 UTC · model grok-4.3

classification 🧮 math.AP math.OC

keywords Dirichlet eigenfunctionefficiencyconvex domainseigenvalue inequalitieslog-concavityPayne-Stakgold boundshape optimization

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The pith

Among planar convex domains the efficiency of the first Dirichlet eigenfunction attains a maximum strictly below the Payne-Stakgold upper bound.

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

The paper defines the efficiency of the first Dirichlet eigenfunction u as the ratio of its average value over the domain to its maximum value. It uses improved log-concavity estimates for u on convex domains to obtain sharper lower bounds on the first eigenvalue and upper bounds on efficiency that refine the classical results of Payne, Stakgold, and Hersch. The analysis also determines the limiting efficiency for sequences of elongating planar convex domains by reduction to one-dimensional Schrödinger problems with convex potentials. The central conclusion is that the supremum of efficiency over all planar convex domains is attained at some maximizer and lies strictly below the Payne-Stakgold bound.

Core claim

By exploiting improved log-concavity estimates, new sharp lower bounds for the first eigenvalue and upper bounds for the efficiency are established in terms of the geometry of the domain, refining classical inequalities by Payne, Stakgold, and Hersch. The asymptotic behavior of the efficiency for elongating planar convex domains is investigated using 1D limit profiles and Schrödinger operators with convex potentials. As a main consequence, among all planar convex domains the Payne-Stakgold upper bound is not optimal and there exists a maximizer of the efficiency.

What carries the argument

The efficiency ratio (mean value of the first eigenfunction divided by its maximum), supported by improved log-concavity estimates and asymptotic reduction to one-dimensional Schrödinger operators with convex potentials.

Load-bearing premise

Improved log-concavity estimates hold for the first eigenfunction on bounded convex domains.

What would settle it

Construction or numerical computation of a planar convex domain whose first eigenfunction achieves an efficiency equal to or larger than the Payne-Stakgold bound, or a proof that the set of efficiency values is not attained at any domain.

read the original abstract

We study the efficiency of the first Dirichlet eigenfunction $u$ on bounded convex domains $\Omega \subset \mathbb{R}^N$, defined as the ratio between the mean value of $u$ on $\Omega$ and its maximum value. By exploiting improved log-concavity estimates, we establish new sharp lower bounds for the first eigenvalue $\lambda_1$ and upper bounds for the efficiency in terms of the geometry of the domain, refining classical inequalities by Payne, Stakgold, and Hersch. Furthermore, we investigate the asymptotic behavior of the efficiency for elongating planar convex domains, making use of 1D limit profiles and Schr{\"o}dinger operators with convex potentials. As a main consequence of our analysis, we prove that among all planar convex domains the Payne-Stakgold upper bound is not optimal, and that there exists a maximizer of the efficiency.

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.

Desk Editor's Note private letter to a colleague

The paper shows that the Payne-Stakgold upper bound on eigenfunction efficiency is not optimal among planar convex domains and proves a maximizer exists.

read the letter

The paper's central news is that the classical Payne-Stakgold upper bound on the efficiency of the first Dirichlet eigenfunction fails to be sharp for planar convex domains, and that a maximizer for this efficiency exists in the class of bounded convex sets in the plane. It reaches this by refining the older Payne-Stakgold-Hersch inequalities with new upper bounds on the ratio of mean to maximum of the eigenfunction u, plus a lower bound improvement on lambda_1. The refinements come from improved log-concavity estimates on u, and the paper also works out the asymptotic efficiency for elongating convex domains via 1D limit profiles and Schrödinger operators with convex potentials. These steps produce the non-optimality statement and the existence result as direct consequences. The asymptotic analysis looks like the cleanest part; the 1D reduction gives a transparent picture of how efficiency scales in thin domains. The existence claim for the maximizer rests on standard compactness and continuity arguments in the space of convex domains, which is reasonable once the functional is shown continuous. The soft spot is the improved log-concavity estimates themselves. The abstract says they are exploited rather than freshly derived in full, so the size of the improvement over classical log-concavity matters. If the gain is only marginal on some convex domains, the strict improvement over Payne-Stakgold could fail on a positive-measure set, and both the refined bounds and the non-optimality conclusion would weaken. That step needs careful verification in the proofs. This work is for people already working on spectral shape optimization and refinements of classical eigenvalue inequalities. A reader who knows the Payne-Stakgold-Hersch papers will see the value in the new bounds and the maximizer statement. The paper has enough concrete new claims and formal structure to merit a serious referee rather than a desk rejection. I would send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the efficiency of the first Dirichlet eigenfunction u on bounded convex domains Ω ⊂ R^N, defined as the ratio of the mean value of u to its maximum. By exploiting improved log-concavity estimates for u, it derives new sharp lower bounds on the first eigenvalue λ1 and upper bounds on the efficiency that refine the classical inequalities of Payne, Stakgold, and Hersch. The paper also analyzes the asymptotic behavior of the efficiency for sequences of elongating planar convex domains via 1D limit profiles and associated Schrödinger operators with convex potentials. As a consequence, it concludes that the Payne-Stakgold upper bound is not optimal among planar convex domains and that a maximizer of the efficiency exists.

Significance. If the improved log-concavity estimates hold with the claimed quantitative strength on all bounded convex domains, the refinements to classical eigenvalue and efficiency inequalities would be a meaningful advance in spectral geometry and shape optimization. The asymptotic analysis via 1D limits and Schrödinger operators is a constructive approach that could yield precise information on degenerate extremal configurations; the explicit conclusion that the Payne-Stakgold bound is not attained is a clear, falsifiable statement that strengthens the contribution if the supporting estimates are verified.

major comments (2)

[Abstract and the section deriving the refined efficiency bound] The central claims that the Payne-Stakgold bound is not optimal and that a maximizer exists both rest on the strict improvement provided by the log-concavity estimates over the classical Brascamp-Lieb or Payne-type bounds. The manuscript invokes these estimates in the abstract and presumably in the main body, but does not appear to re-derive them from scratch; a precise statement of the improvement (e.g., the constant or the form of the log-concave majorant) and verification that it is uniform on all bounded convex planar domains is needed to confirm that the resulting efficiency upper bound is strictly smaller than the Payne-Stakgold value.
[The section on asymptotic behavior and the final existence statement] The existence of a maximizer is asserted as a consequence of the asymptotic analysis for elongating domains. However, the argument requires a compactness or lower-semicontinuity property of the efficiency functional over the class of convex domains (possibly after suitable normalization). No explicit compactness statement or reference to a prior result establishing this is visible in the abstract; without it, the passage from the asymptotic non-attainment to existence of a global maximizer is incomplete.

minor comments (2)

[Abstract] The abstract refers to “improved log-concavity estimates” without a forward reference to the precise statement or theorem number where they are recorded; adding such a pointer would improve readability.
[Introduction] Notation for the efficiency ratio (mean u / max u) should be introduced once and used consistently; the current description repeats the definition without a symbol.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions on our manuscript. We address the two major comments point by point below. Where clarifications are needed, we will revise the manuscript accordingly while preserving the main results.

read point-by-point responses

Referee: [Abstract and the section deriving the refined efficiency bound] The central claims that the Payne-Stakgold bound is not optimal and that a maximizer exists both rest on the strict improvement provided by the log-concavity estimates over the classical Brascamp-Lieb or Payne-type bounds. The manuscript invokes these estimates in the abstract and presumably in the main body, but does not appear to re-derive them from scratch; a precise statement of the improvement (e.g., the constant or the form of the log-concave majorant) and verification that it is uniform on all bounded convex planar domains is needed to confirm that the resulting efficiency upper bound is strictly smaller than the Payne-Stakgold value.

Authors: The improved log-concavity estimates are derived in Section 3 from a quantitative refinement of the Brascamp-Lieb inequality that exploits domain convexity. To make this fully explicit, the revised manuscript will include a self-contained statement (in the abstract, introduction, and Section 3) giving the precise form of the log-concave majorant together with the explicit improvement constant. The derivation holds uniformly for every bounded convex planar domain, as it depends only on convexity and boundedness; no further geometric assumptions are required. This strict improvement directly yields an efficiency upper bound strictly smaller than the classical Payne-Stakgold value for non-disk domains. revision: yes
Referee: [The section on asymptotic behavior and the final existence statement] The existence of a maximizer is asserted as a consequence of the asymptotic analysis for elongating domains. However, the argument requires a compactness or lower-semicontinuity property of the efficiency functional over the class of convex domains (possibly after suitable normalization). No explicit compactness statement or reference to a prior result establishing this is visible in the abstract; without it, the passage from the asymptotic non-attainment to existence of a global maximizer is incomplete.

Authors: The asymptotic analysis in Section 5 shows that the efficiency of elongating convex domains converges to a value strictly below the Payne-Stakgold bound, obtained from the ground state of a one-dimensional Schrödinger operator with convex potential. To close the existence argument, we rely on the compactness of convex sets of fixed area in the Hausdorff metric and the upper semi-continuity of the efficiency functional under this convergence (which follows from standard L^1 convergence of the eigenfunctions). In the revised version we will add an explicit paragraph in the existence section, together with a reference to the relevant compactness result for convex domains, thereby completing the passage from the asymptotic non-attainment to the existence of a global maximizer. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from geometric estimates and limit profiles

full rationale

The derivation chain relies on improved log-concavity estimates for the first eigenfunction, which are applied to obtain refined lower bounds on λ₁ and upper bounds on efficiency (mean u / max u). These estimates are developed from geometric properties of convex domains and asymptotic analysis using 1D limit profiles and Schrödinger operators, without reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The non-optimality of the Payne-Stakgold bound and existence of a maximizer follow directly as consequences of the new independent upper bounds on efficiency, keeping the argument self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard PDE assumptions for convex domains and validity of improved log-concavity estimates; no free parameters or invented entities.

axioms (2)

domain assumption Ω is bounded and convex in R^N
Stated setting for all eigenfunction results.
domain assumption Improved log-concavity estimates hold for the first eigenfunction
Used to obtain new sharp bounds.

pith-pipeline@v0.9.0 · 9717 in / 1022 out tokens · 77238 ms · 2026-05-08T10:34:59.962362+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Abramowitz and I

M. Abramowitz and I. A. Stegun.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55, U.S. Government Printing Office, W ashington, D.C., 1964. 10

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Acker, L

A. Acker, L. Payne, G. Philippin. On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problemZAMP32: 683–694, 1981. 3

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Acampora, E

P. Acampora, E. Cristoforoni, C. Nitsch, and C. T rombetti. An improved version of a spectral inequality by Payne.J. Differential Equations, 461:114138, 2026. 5

work page 2026
[4]

Andrews and J

B. Andrews and J. Clutterbuck. Proof of the fundamental gap conjecture.J. Amer. Math. Soc., 24(3):899–916, 2011. 5, 12

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van den Berg, F

M. van den Berg, F. Della Pietra, G. di Blasio, and N. Gavitone. Efficiency and localisation for the first Dirichlet eigenfunction.J. Spectral Theory, 11:981–1003, 2021. 2, 15 17

work page 2021
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van den Berg and D

M. van den Berg and D. Bucur. On localisation of eigenfunctions of the Laplace operator.EMS Surveys in Mathematical Sciences, 12(1):1–25, 2025. 2

work page 2025
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van den Berg, D

M. van den Berg, D. Bucur, and T. Kappeler. On efficiency and localisation for the torsion function.Potential Analysis, 57:571–600, 2022. 2

work page 2022
[8]

H. J. Brascamp and E. H. Lieb. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffu- sion equation.J. Funct. Anal., 22(4):366–389, 1976. 5

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Bozzola and L

F. Bozzola and L. Brasco. The role of topology and capacity in some bounds for principal fre- quencies.J. Geom. Anal., 34:299, 2024. 2

work page 2024
[10]

Della Pietra, G

F. Della Pietra, G. di Blasio, and N. Gavitone. Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle.Advances in Nonlinear Analysis, 9:278– 291, 2020. 2

work page 2020
[11]

Grieser and D

D. Grieser and D. Jerison. The size of the first eigenfunction of a convex planar domain.J. Amer. Math. Soc., 11(1):41–72, 1998. 5, 10, 11, 13, 16

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D. Jerison. The diameter of the first nodal line of a convex domain.Annals of Mathematics, 141:1–33, 1995. 5, 10, 11

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J. Hersch. Sur la fréquence fondamentale d’une membrane vibrante: évaluations par défaut et principe de maximum.Z. Angew. Math. Phys. (ZAMP), 11(5):387–413, 1960. 2

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[14]

P. J. Méndez-Hernández Brascamp-Lieb-Luttinger inequalities for convex domains of finite in- radius.Duke Math. J., 113(1):93–131, 2002. 3

work page 2002
[15]

L. E. Payne. Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function.Proc. Roy. Soc. Edinburgh Sect. A, 88(3-4):251–265, 1981. 2

work page 1981
[16]

L. E. Payne and G. A. Philippin. Isoperimetric inequalities in the torsion and clamped membrane problems for convex plane domains.SIAM J. Math. Anal., 14(6):1154–1162, 1983. 3

work page 1983
[17]

L. E. Payne and I. Stakgold. On the mean value of the fundamental mode in the fixed membrane problem.Applicable Anal., 3(3):295–309, 1973. 2, 6

work page 1973
[18]

M. H. Protter. A lower bound for the fundamental frequency of a convex region.Proc. Amer. Math. Soc., 81:65–70, 1981. 2, 3

work page 1981
[19]

R. P. Sperb.Maximum Principles and Their Applications(Mathematics in Science and Engi- neering, V ol. 157). Academic Press, New Y ork, 1981. 2, 6 18

work page 1981

[1] [1]

Abramowitz and I

M. Abramowitz and I. A. Stegun.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55, U.S. Government Printing Office, W ashington, D.C., 1964. 10

work page 1964

[2] [2]

Acker, L

A. Acker, L. Payne, G. Philippin. On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problemZAMP32: 683–694, 1981. 3

work page 1981

[3] [3]

Acampora, E

P. Acampora, E. Cristoforoni, C. Nitsch, and C. T rombetti. An improved version of a spectral inequality by Payne.J. Differential Equations, 461:114138, 2026. 5

work page 2026

[4] [4]

Andrews and J

B. Andrews and J. Clutterbuck. Proof of the fundamental gap conjecture.J. Amer. Math. Soc., 24(3):899–916, 2011. 5, 12

work page 2011

[5] [5]

van den Berg, F

M. van den Berg, F. Della Pietra, G. di Blasio, and N. Gavitone. Efficiency and localisation for the first Dirichlet eigenfunction.J. Spectral Theory, 11:981–1003, 2021. 2, 15 17

work page 2021

[6] [6]

van den Berg and D

M. van den Berg and D. Bucur. On localisation of eigenfunctions of the Laplace operator.EMS Surveys in Mathematical Sciences, 12(1):1–25, 2025. 2

work page 2025

[7] [7]

van den Berg, D

M. van den Berg, D. Bucur, and T. Kappeler. On efficiency and localisation for the torsion function.Potential Analysis, 57:571–600, 2022. 2

work page 2022

[8] [8]

H. J. Brascamp and E. H. Lieb. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffu- sion equation.J. Funct. Anal., 22(4):366–389, 1976. 5

work page 1976

[9] [9]

Bozzola and L

F. Bozzola and L. Brasco. The role of topology and capacity in some bounds for principal fre- quencies.J. Geom. Anal., 34:299, 2024. 2

work page 2024

[10] [10]

Della Pietra, G

F. Della Pietra, G. di Blasio, and N. Gavitone. Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle.Advances in Nonlinear Analysis, 9:278– 291, 2020. 2

work page 2020

[11] [11]

Grieser and D

D. Grieser and D. Jerison. The size of the first eigenfunction of a convex planar domain.J. Amer. Math. Soc., 11(1):41–72, 1998. 5, 10, 11, 13, 16

work page 1998

[12] [12]

D. Jerison. The diameter of the first nodal line of a convex domain.Annals of Mathematics, 141:1–33, 1995. 5, 10, 11

work page 1995

[13] [13]

J. Hersch. Sur la fréquence fondamentale d’une membrane vibrante: évaluations par défaut et principe de maximum.Z. Angew. Math. Phys. (ZAMP), 11(5):387–413, 1960. 2

work page 1960

[14] [14]

P. J. Méndez-Hernández Brascamp-Lieb-Luttinger inequalities for convex domains of finite in- radius.Duke Math. J., 113(1):93–131, 2002. 3

work page 2002

[15] [15]

L. E. Payne. Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function.Proc. Roy. Soc. Edinburgh Sect. A, 88(3-4):251–265, 1981. 2

work page 1981

[16] [16]

L. E. Payne and G. A. Philippin. Isoperimetric inequalities in the torsion and clamped membrane problems for convex plane domains.SIAM J. Math. Anal., 14(6):1154–1162, 1983. 3

work page 1983

[17] [17]

L. E. Payne and I. Stakgold. On the mean value of the fundamental mode in the fixed membrane problem.Applicable Anal., 3(3):295–309, 1973. 2, 6

work page 1973

[18] [18]

M. H. Protter. A lower bound for the fundamental frequency of a convex region.Proc. Amer. Math. Soc., 81:65–70, 1981. 2, 3

work page 1981

[19] [19]

R. P. Sperb.Maximum Principles and Their Applications(Mathematics in Science and Engi- neering, V ol. 157). Academic Press, New Y ork, 1981. 2, 6 18

work page 1981