Sharp quantitative Talenti's inequality in particular cases

Jimmy Lamboley; Paolo Acampora

arxiv: 2503.07337 · v2 · submitted 2025-03-10 · 🧮 math.AP

Sharp quantitative Talenti's inequality in particular cases

Paolo Acampora , Jimmy Lamboley This is my paper

Pith reviewed 2026-05-23 00:52 UTC · model grok-4.3

classification 🧮 math.AP

keywords Talenti inequalitySchwarz symmetrizationstability estimatesL^p normscharacteristic functionsunit ballDirichlet boundary conditionsquantitative inequalities

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The pith

For right-hand sides that are characteristic functions inside the unit ball, the L^p Talenti inequality admits stability with the sharp exponent 2.

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

The paper examines Talenti's symmetrization inequality in its L^p form, comparing the L^p norms of solutions to Poisson equations driven by a nonnegative source and by its Schwarz symmetrization. Attention is limited to the unit ball as the domain and to sources that are characteristic functions of arbitrary subsets of that ball, all under Dirichlet boundary conditions. In this restricted setting the authors prove a quantitative stability statement: the deficit between the two L^p norms is bounded from below by a positive multiple of the square of a measure of asymmetry. This identifies the exponent 2 as achievable and sharp for these data. A reader cares because the result gives an explicit rate at which equality is approached when the source set approaches radial symmetry.

Core claim

When the right-hand side f is the characteristic function of an arbitrary subset of the unit ball, the L^p norm of the solution to -Δv = f^♯ exceeds the L^p norm of the solution to -Δu = f by an amount at least a positive constant times the square of a suitable measure of how far the set is from being a centered ball.

What carries the argument

The quantitative stability estimate with exponent 2 relating the L^p deficit to the squared asymmetry of the support of f under Dirichlet conditions on the unit ball.

Load-bearing premise

The right-hand side f is the characteristic function of an arbitrary subset of the unit ball (with Dirichlet boundary conditions on the unit ball).

What would settle it

A sequence of subsets of the unit ball whose asymmetry tends to zero while the ratio of L^p deficit to the square of asymmetry tends to zero would show that exponent 2 is not achieved.

read the original abstract

In this paper, we focus on the famous Talenti's symmetrization inequality, more precisely its $L^p$ corollary asserting that the $L^p$-norm of the solution to $-\Delta v=f^\sharp$ is higher than the $L^p$-norm of the solution to $-\Delta u=f$ (we are considering Dirichlet boundary conditions, and $f^\sharp$ denotes the Schwarz symmetrization of $f:\Omega\to\mathbb{R}_+$). We focus on the particular case where functions $f$ are defined on the unit ball, and are characteristic functions of a subset of this unit ball. We show in this case that stability occurs for the $L^p$-Talenti inequality with the sharp exponent 2.

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

This paper proves stability with the sharp exponent 2 for the L^p-Talenti inequality exactly when f is the characteristic function of an arbitrary subset of the unit ball.

read the letter

The core contribution is a stability statement for the L^p version of Talenti's inequality that holds with the optimal power 2 in one tightly scoped case: f is the indicator of any subset of the unit ball, Dirichlet conditions on the ball itself. The abstract states the result cleanly and does not overclaim generality, which is a plus. They isolate the setting where the sharp constant is attainable and deliver it without extra regularity assumptions on the subset. That is honest, incremental progress inside the symmetrization literature. The work is scoped precisely enough that the claim does not appear to rest on fitting or circular reasoning. The restriction to characteristic functions on the ball is explicit, so the result stands or falls on its own terms in that class. The main limitation is the narrowness. The paper does not address general positive f, other domains, or non-characteristic data, so the stability exponent 2 is not shown to carry over. That is not a defect in the argument but it does cap the immediate reach. Without the full proof one cannot check the technical steps, yet the stress-test note finds no internal inconsistency in the stated claim, and the abstract gives no sign of hidden fitting or unsupported jumps. Specialists working on quantitative rearrangement inequalities or stability in elliptic problems will find this useful as a precise data point. A reader outside that subfield will not get much. The paper is coherent on its own terms and the result is new in the claimed setting, so it merits sending to referees for a proper check of the argument.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a sharp quantitative stability result for the L^p version of Talenti's symmetrization inequality (comparing ||v||_p and ||u||_p for solutions of -Δv = f^♯ and -Δu = f with Dirichlet conditions). The result is proved in the restricted setting where the domain is the unit ball and f is the characteristic function of an arbitrary subset of that ball; the authors claim that stability holds with the sharp exponent 2.

Significance. If the proof is correct, the result supplies an explicit, sharp stability exponent in a concrete but nontrivial special case of the quantitative Talenti inequality. This supplies a benchmark that can be used to test conjectures about the general exponent and may be useful in applications of rearrangement techniques to elliptic problems on balls.

minor comments (3)

[Abstract] The abstract and introduction should state the precise range of p>1 for which the stability result holds (e.g., whether it is all p or only p in a subinterval).
[Introduction] Notation for the Schwarz symmetrization f^♯ and the solutions u,v should be introduced with a short reminder of the underlying PDE in the first paragraph of the introduction.
A brief remark on whether the arbitrary subset may be taken measurable only or requires additional regularity would clarify the scope of the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the sharp quantitative stability for the L^p-Talenti inequality in the special case of characteristic functions on the unit ball. The recommendation of minor revision is noted. However, the report lists no specific major comments, so we have no individual points requiring point-by-point rebuttal or revision at this stage. The result as claimed in the abstract stands.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a theorem asserting stability with sharp exponent 2 for the L^p-Talenti inequality precisely when f is the characteristic function of an arbitrary subset of the unit ball (Dirichlet conditions on the ball). The provided abstract and description frame this as the conclusion of a proof, with no equations, self-citations, or steps that reduce the claimed stability constant or exponent to a fitted input or prior self-referential definition. The restriction to this function class is explicit and the result is scoped exactly to it, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the standard properties of Schwarz symmetrization and the classical Talenti comparison; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Schwarz symmetrization of a nonnegative function preserves the L^1 norm and yields a radially decreasing function whose superlevel sets are balls.
This is the definition underlying the comparison between u and v in Talenti's inequality.
standard math The solution operator for the Dirichlet Laplacian on the unit ball maps L^1 data to L^p functions for p in the appropriate range.
Used implicitly when comparing L^p norms of the two solutions.

pith-pipeline@v0.9.0 · 5641 in / 1332 out tokens · 36945 ms · 2026-05-23T00:52:30.880582+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 … ∥u_{B^*}∥_{L^p(B_1)} − ∥u_E∥_{L^p(B_1)} ≥ c |E Δ B^*|^2 … proved via shape derivatives of the Lagrangian L_τ(E) = J(E) + τ|E| and coercivity of L''_τ(B^*)[Φ,Φ] ≤ −c ∥Φ·ν_0∥^2_{L^2(∂B^*)}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 4.12 … spherical harmonics Y_{k,m} are eigenfunctions of T with eigenvalues 1/λ_k … λ_1 > (|∂_ν w_0||_{∂B^*})^{-1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Quantitative Inequalities for the Expecte d Lifetime of Brownian Motion

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Stability in shape optimi zation with second variation

M. Dambrine and J. Lamboley. “Stability in shape optimi zation with second variation”. In: J. Diﬀerential Equations 267.5 (2019), pp. 3009–3045. issn: 0022-0396,1090-2732. doi: 10.1016/j.jde.2019.03.033 . url: https://doi.org/10.1016/j.jde.2019.03. 033

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A quantitative form of Fabe r–Krahn inequality

N. Fusco and Y. R.-Y. Zhang. “A quantitative form of Fabe r–Krahn inequality”. en. In: Calculus of Variations and Partial Diﬀerential Equations 56.5 (Aug. 2017), p. 138. issn: 1432-0835. doi: 10.1007/s00526-017-1224-7 . url: https://doi.org/10.1007/ s00526-017-1224-7 (visited on 02/23/2025)

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Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial diﬀerential equations of second order . English. Reprint of the 1998 ed. Class. Math. Berlin: Springe r, 2001. isbn: 3-540-41160- 7. 43

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On two funct ionals involving the maximum of the torsion function

A. Henrot, I. Lucardesi, and G. Philippin. “On two funct ionals involving the maximum of the torsion function”. en. In: ESAIM: Control, Optimisation and Calculus of Varia- tions 24.4 (Oct. 2018). Number: 4 Publisher: EDP Sciences, pp. 158 5–1604. issn: 1292- 8119, 1262-3377. doi: 10.1051/cocv/2017069 . url: https://www.esaim-cocv.org/ articles/cocv/abs/20...

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Henrot and M

A. Henrot and M. Pierre. Shape variation and optimization. A geometrical analysis . English. Vol. 28. EMS Tracts Math. Zürich: European Mathema tical Society (EMS),

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S. Kesavan. Symmetrization & applications . eng. Series in analysis ; v. 3. Hackensack, N.J: World Scientiﬁc, 2006. isbn: 981256733X

work page 2006
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Estimates of ﬁr st and second order shape derivatives in nonsmooth multidimensional domains and app lications

J. Lamboley, A. Novruzi, and M. Pierre. “Estimates of ﬁr st and second order shape derivatives in nonsmooth multidimensional domains and app lications”. In: J. Funct. Anal. 270.7 (2016), pp. 2616–2652. issn: 0022-1236,1096-0783. doi: 10.1016/j.jfa.2016.02.013. url: https://doi.org/10.1016/j.jfa.2016.02.013

work page doi:10.1016/j.jfa.2016.02.013 2016
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Quantitative inequality for the eigenvalu e of a Schrödinger operator in the ball

I. Mazari. “Quantitative inequality for the eigenvalu e of a Schrödinger operator in the ball”. In: Journal of Diﬀerential Equations 269.11 (2020), pp. 10181–10238. issn: 0022-

work page 2020
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doi: https://doi.org/10.1016/j.jde.2020.06.057

work page doi:10.1016/j.jde.2020.06.057 2020
[21]

Quantitative Stability fo r Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem

I. Mazari and D. Ruiz-Balet. “Quantitative Stability fo r Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem”. In: SIAM Journal on Mathematical Analysis 54.3 (2022), pp. 3848–3883. doi: 10.1137/21M1393121

work page doi:10.1137/21m1393121 2022
[22]

Fuglede-Type Arguments for Isoperimetri c Problems and Applications to Stability Among Convex Shapes

R. Prunier. “Fuglede-Type Arguments for Isoperimetri c Problems and Applications to Stability Among Convex Shapes”. In: SIAM Journal on Mathematical Analysis 56.2 (2024), pp. 1560–1603. doi: 10.1137/23M1567412 . url: https://doi.org/10.1137/ 23M1567412

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Elliptic equations and rearrangements

G. Talenti. “Elliptic equations and rearrangements”. English. In: Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 3 (1976), pp. 697–718. issn: 0391-173X

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Inequalities in rearrangement invariant function spaces

G. Talenti. “Inequalities in rearrangement invariant function spaces”. English. In: Non- linear analysis, function spaces and applications. Vol. 5. Proceedings of the spring school held in Prague, May 23-28, 1994 . Prague: Prometheus Publishing House, 1994, pp. 177–

work page 1994
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isbn: 80-85849-69-0. 44

work page

[1] [1]

Inves tigating optimal and quanti- tative versions of the Talenti’s inequality

P. Acampora, J. Lamboley, and I. Mazari-Fouquer. “Inves tigating optimal and quanti- tative versions of the Talenti’s inequality”. In: (in preparation) (2025)

work page 2025

[2] [2]

A remark on comp arison results via sym- metrization

A. Alvino, P.L. Lions, and G. Trombetti. “A remark on comp arison results via sym- metrization”. In: Proceedings of the Royal Society of Edinburgh: Section A Mat hematics 102.1–2 (1986), pp. 37–48. doi: 10.1017/S0308210500014475

work page doi:10.1017/s0308210500014475 1986

[3] [3]

The Tal enti comparison result in a quantitative form

V. Amato, R. Barbato, A. L. Masiello, and G. Paoli. “The Tal enti comparison result in a quantitative form”. In: Annali Scuola Normale Superiore - Classe Di Scienze 31 (2024). doi: https://doi.org/10.2422/2036-2145.202404_010

work page doi:10.2422/2036-2145.202404_010 2024

[4] [4]

H. Cartan. Calcul diﬀérentiel. French. Collection méthodes. Paris: Hermann & Cie. 178 p. (1967). 1967

work page 1967

[5] [5]

Stabil ity of optimal shapes and con- vergence of thresholding algorithms in linear and spectral optimal control problems

A. Chambolle, I. Mazari-Fouquer, and Y. Privat. “Stabil ity of optimal shapes and con- vergence of thresholding algorithms in linear and spectral optimal control problems”. working paper or preprint. June 2023. url: https://hal.science/hal-04140177

work page 2023

[6] [6]

Stabil ity of optimal shapes and con- vergence of thresholding algorithms in linear and spectral optimal control problems: Supplementary material

A. Chambolle, I. Mazari-Fouquer, and Y. Privat. “Stabil ity of optimal shapes and con- vergence of thresholding algorithms in linear and spectral optimal control problems: Supplementary material”. working paper or preprint. June 2 023. url: https : //hal. science/hal-04140334

work page

[7] [7]

Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Comp osite Membranes

S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnish i. “Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Comp osite Membranes”. en. In: Communications in Mathematical Physics 214.2 (Nov. 2000), pp. 315–337. issn: 1432-0916. doi: 10.1007/PL00005534 . url: https://doi.org/10.1007/PL00005534 (visited on 02/12/2025)

work page doi:10.1007/pl00005534 2000

[8] [8]

A Selection Principle fo r the Sharp Quantitative Isoperimetric Inequality

M. Cicalese and G. P. Leonardi. “A Selection Principle fo r the Sharp Quantitative Isoperimetric Inequality”. In: Archive for Rational Mechanics and Analysis 206.2 (2012), pp. 617–643. doi: 10.1007/s00205- 012- 0544- 1 . url: https://doi.org/10.1007/ s00205-012-0544-1

work page doi:10.1007/s00205- 2012

[9] [9]

Quantitative Inequalities for the Expecte d Lifetime of Brownian Motion

K. Daesung. “Quantitative Inequalities for the Expecte d Lifetime of Brownian Motion”. In: Michigan Mathematical Journal 70.3 (2021), pp. 615–634. doi: 10.1307/mmj/1593136867. url: https://doi.org/10.1307/mmj/1593136867

work page doi:10.1307/mmj/1593136867 2021

[10] [10]

Stability in shape optimi zation with second variation

M. Dambrine and J. Lamboley. “Stability in shape optimi zation with second variation”. In: J. Diﬀerential Equations 267.5 (2019), pp. 3009–3045. issn: 0022-0396,1090-2732. doi: 10.1016/j.jde.2019.03.033 . url: https://doi.org/10.1016/j.jde.2019.03. 033

work page doi:10.1016/j.jde.2019.03.033 2019

[11] [11]

Stability in the Isoperimetric Problem f or Convex or Nearly Spherical Domains in Rn

Bent Fuglede. “Stability in the Isoperimetric Problem f or Convex or Nearly Spherical Domains in Rn”. In: Transactions of the American Mathematical Society 314.2 (1989), pp. 619–638. issn: 00029947, 10886850. url: http://www.jstor.org/stable/2001401 (visited on 10/09/2024)

work page arXiv 1989

[12] [12]

A quantitative form of Fabe r–Krahn inequality

N. Fusco and Y. R.-Y. Zhang. “A quantitative form of Fabe r–Krahn inequality”. en. In: Calculus of Variations and Partial Diﬀerential Equations 56.5 (Aug. 2017), p. 138. issn: 1432-0835. doi: 10.1007/s00526-017-1224-7 . url: https://doi.org/10.1007/ s00526-017-1224-7 (visited on 02/23/2025)

work page doi:10.1007/s00526-017-1224-7 2017

[13] [13]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial diﬀerential equations of second order . English. Reprint of the 1998 ed. Class. Math. Berlin: Springe r, 2001. isbn: 3-540-41160- 7. 43

work page 1998

[14] [14]

On two funct ionals involving the maximum of the torsion function

A. Henrot, I. Lucardesi, and G. Philippin. “On two funct ionals involving the maximum of the torsion function”. en. In: ESAIM: Control, Optimisation and Calculus of Varia- tions 24.4 (Oct. 2018). Number: 4 Publisher: EDP Sciences, pp. 158 5–1604. issn: 1292- 8119, 1262-3377. doi: 10.1051/cocv/2017069 . url: https://www.esaim-cocv.org/ articles/cocv/abs/20...

work page doi:10.1051/cocv/2017069 2018

[15] [15]

Henrot and M

A. Henrot and M. Pierre. Shape variation and optimization. A geometrical analysis . English. Vol. 28. EMS Tracts Math. Zürich: European Mathema tical Society (EMS),

work page

[16] [16]

doi: 10.4171/178

isbn: 978-3-03719-178-1; 978-3-03719-678-6. doi: 10.4171/178

work page doi:10.4171/178

[17] [17]

S. Kesavan. Symmetrization & applications . eng. Series in analysis ; v. 3. Hackensack, N.J: World Scientiﬁc, 2006. isbn: 981256733X

work page 2006

[18] [18]

Estimates of ﬁr st and second order shape derivatives in nonsmooth multidimensional domains and app lications

J. Lamboley, A. Novruzi, and M. Pierre. “Estimates of ﬁr st and second order shape derivatives in nonsmooth multidimensional domains and app lications”. In: J. Funct. Anal. 270.7 (2016), pp. 2616–2652. issn: 0022-1236,1096-0783. doi: 10.1016/j.jfa.2016.02.013. url: https://doi.org/10.1016/j.jfa.2016.02.013

work page doi:10.1016/j.jfa.2016.02.013 2016

[19] [19]

Quantitative inequality for the eigenvalu e of a Schrödinger operator in the ball

I. Mazari. “Quantitative inequality for the eigenvalu e of a Schrödinger operator in the ball”. In: Journal of Diﬀerential Equations 269.11 (2020), pp. 10181–10238. issn: 0022-

work page 2020

[20] [20]

doi: https://doi.org/10.1016/j.jde.2020.06.057

work page doi:10.1016/j.jde.2020.06.057 2020

[21] [21]

Quantitative Stability fo r Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem

I. Mazari and D. Ruiz-Balet. “Quantitative Stability fo r Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem”. In: SIAM Journal on Mathematical Analysis 54.3 (2022), pp. 3848–3883. doi: 10.1137/21M1393121

work page doi:10.1137/21m1393121 2022

[22] [22]

Fuglede-Type Arguments for Isoperimetri c Problems and Applications to Stability Among Convex Shapes

R. Prunier. “Fuglede-Type Arguments for Isoperimetri c Problems and Applications to Stability Among Convex Shapes”. In: SIAM Journal on Mathematical Analysis 56.2 (2024), pp. 1560–1603. doi: 10.1137/23M1567412 . url: https://doi.org/10.1137/ 23M1567412

work page doi:10.1137/23m1567412 2024

[23] [23]

Elliptic equations and rearrangements

G. Talenti. “Elliptic equations and rearrangements”. English. In: Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 3 (1976), pp. 697–718. issn: 0391-173X

work page 1976

[24] [24]

Inequalities in rearrangement invariant function spaces

G. Talenti. “Inequalities in rearrangement invariant function spaces”. English. In: Non- linear analysis, function spaces and applications. Vol. 5. Proceedings of the spring school held in Prague, May 23-28, 1994 . Prague: Prometheus Publishing House, 1994, pp. 177–

work page 1994

[25] [25]

isbn: 80-85849-69-0. 44

work page