The equilibrium measure for an anisotropic nonlocal energy

J. A. Carrillo; J. Mateu; J. Verdera; L. Rondi; L. Scardia; M. G. Mora

arxiv: 1907.00417 · v1 · pith:VDEDRSLFnew · submitted 2019-06-30 · 🧮 math.AP

The equilibrium measure for an anisotropic nonlocal energy

J. A. Carrillo , J. Mateu , M. G. Mora , L. Rondi , L. Scardia , J. Verdera This is my paper

Pith reviewed 2026-05-25 12:26 UTC · model grok-4.3

classification 🧮 math.AP

keywords nonlocal energyanisotropic kernelequilibrium measurespheroidminimiserFourier transformvariational problem

0 comments

The pith

For alpha in (-1, n-2] the unique minimiser of the anisotropic nonlocal energy is the characteristic function of a spheroid.

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

The paper proves that in dimensions n at least 3 the minimisers of a family of anisotropic nonlocal energies with quadratic confinement are unique and equal to the normalised characteristic functions of spheroids for anisotropy parameters alpha between -1 and n-2. This extends two-dimensional results by showing that loss of dimensionality does not occur at the critical alpha equal to n-2. A reader would care because the result illustrates how kernel anisotropy determines the geometry of equilibrium measures in variational problems.

Core claim

We prove that for α∈(-1,n−2], the minimiser of I_α is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension n=2, does not occur in higher dimension at the value α=n-2 corresponding to the sign change of the Fourier transform of the interaction potential.

What carries the argument

The Fourier transform of the interaction kernel, whose positivity for alpha less than n-2 and sign change at alpha equals n-2 keeps the energy quadratic form positive definite and prevents dimensionality loss.

If this is right

The minimiser is unique for all alpha in the given interval.
The support of the minimiser is always a spheroid.
Loss of dimensionality does not occur at alpha equals n-2 in dimensions three and higher.
The result applies specifically when the kernel reduces to the Coulomb potential at alpha equals zero.

Where Pith is reading between the lines

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

The anisotropy in higher dimensions is strong enough to maintain full support even when the kernel's Fourier transform changes sign.
Similar results might hold for other forms of confinement if the quadratic form positivity can be verified.
Testing the minimiser shape numerically for specific n and alpha values could provide independent confirmation.

Load-bearing premise

The quadratic form of the energy stays positive definite on probability measures even at the alpha value where the Fourier transform of the kernel changes sign.

What would settle it

A numerical or analytical example of a non-spheroidal minimiser or multiple minimisers for some alpha in (-1, n-2] would disprove the uniqueness and shape claim.

read the original abstract

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $I_\alpha$ defined on probability measures in $\R^n$, with $n\geq 3$. The energy $I_\alpha$ consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $\alpha=0$ and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for $\alpha\in (-1, n-2]$, the minimiser of $I_\alpha$ is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension $n=2$, does not occur in higher dimension at the value $\alpha=n-2$ corresponding to the sign change of the Fourier transform of the interaction potential.

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 extends the 2D characterization to n≥3 by showing the minimizer is always a spheroid up to α=n-2, with no dimensionality collapse at the Fourier sign-change point.

read the letter

The new content is the explicit geometric description for dimensions three and higher. The 2D case was already done by the same group with complex analysis; here they handle n≥3 with different tools and prove that the unique minimizer of I_α is the normalized characteristic function of a spheroid for every α in (-1, n-2]. That rules out lower-dimensional supports even at the endpoint where the Fourier symbol of the kernel changes sign, which is the main technical step beyond the earlier work. The result is clean and directly addresses how anisotropy affects the shape of the equilibrium measure under quadratic confinement. The abstract states the claim precisely and the circularity burden looks low. The soft spot is exactly the boundary case α = n-2. Positivity of the Fourier transform gives strict convexity for smaller α and immediately excludes competitors, but at the sign-change value that argument stops working, so the proof must rely on a separate comparison or Euler-Lagrange analysis to keep the support full-dimensional. That step is the one that needs the closest look; if it holds, the whole statement is solid. This paper is for researchers in nonlocal variational problems and equilibrium measures who care about anisotropy in higher dimensions. It is a concrete, falsifiable characterization that was missing, so it deserves referee time even if revisions are needed on the endpoint argument.

Referee Report

1 major / 0 minor

Summary. The paper characterizes minimizers of the one-parameter family of nonlocal anisotropic energies I_α (nonlocal convolution term plus quadratic confinement) on probability measures in R^n, n≥3. The central claim is that for every α∈(-1,n-2] the unique minimizer is the normalized characteristic function of a spheroid; the result is presented as an extension of the authors’ earlier two-dimensional work and as evidence that anisotropy prevents loss of dimensionality at the critical value α=n-2 where the Fourier symbol of the kernel changes sign.

Significance. If the proofs are complete, the explicit characterization supplies a rare benchmark example in which the shape of the equilibrium measure is fully determined by the anisotropy of the kernel. The fact that the minimizer remains full-dimensional at the sign-change point distinguishes the higher-dimensional setting from the two-dimensional case and may inform models of defects or aggregation with anisotropic interactions.

major comments (1)

[proof of the main result at the endpoint α=n-2] The argument for α=n-2 must rely on a mechanism other than strict positivity of the Fourier symbol, since the symbol changes sign exactly at this endpoint and the quadratic form ceases to be positive definite. The manuscript should isolate this case (e.g., in the proof of the main theorem or in a dedicated comparison lemma) and verify that every competitor supported on a lower-dimensional set has strictly higher energy; without an explicit estimate or direct comparison at this boundary value the uniqueness claim is not yet load-bearing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit treatment of the endpoint α = n-2. We address the comment below.

read point-by-point responses

Referee: [proof of the main result at the endpoint α=n-2] The argument for α=n-2 must rely on a mechanism other than strict positivity of the Fourier symbol, since the symbol changes sign exactly at this endpoint and the quadratic form ceases to be positive definite. The manuscript should isolate this case (e.g., in the proof of the main theorem or in a dedicated comparison lemma) and verify that every competitor supported on a lower-dimensional set has strictly higher energy; without an explicit estimate or direct comparison at this boundary value the uniqueness claim is not yet load-bearing.

Authors: We agree that the case α = n-2 requires separate handling, since the Fourier symbol of the kernel is no longer strictly positive. The manuscript currently invokes positivity for α < n-2 and passes to the limit at the endpoint, but this does not furnish an explicit energy comparison against lower-dimensional competitors at α = n-2. In the revised manuscript we will isolate the endpoint by inserting a dedicated comparison lemma that directly shows any measure supported on a set of Hausdorff dimension less than n has strictly higher I_{n-2}-energy than the normalized characteristic function of the spheroid. The lemma will exploit the explicit expression of the kernel at α = n-2 together with a direct computation on lower-dimensional subspaces. revision: yes

Circularity Check

0 steps flagged

No circularity: uniqueness follows from kernel Fourier positivity and direct variational comparison, independent of self-citations

full rationale

The paper establishes uniqueness of the spheroid minimizer for α ∈ (-1, n-2] via positivity of the Fourier transform of the kernel for α < n-2 (ensuring strict convexity of the nonlocal term) and a separate argument at the endpoint α = n-2 to rule out lower-dimensional competitors. No quoted step reduces a prediction to a fitted parameter, renames a known result, or loads the central claim on a self-citation whose content is itself unverified. The 2D case is cited only for context and uses different techniques; the n ≥ 3 result is self-contained against the stated assumptions on the kernel and confinement. This is the normal non-circular outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard existence and lower-semicontinuity results for nonlocal energies with Riesz-type kernels, plus the sign properties of the Fourier transform of the anisotropic kernel; no new entities are introduced and no parameters are fitted to data.

axioms (2)

domain assumption The nonlocal energy functional is lower semicontinuous with respect to weak convergence of measures and admits minimizers.
Standard background assumption invoked for any minimization problem on probability measures; location implicit in the statement that a minimiser exists and is unique.
domain assumption The Fourier transform of the interaction kernel is positive definite for α < n-2 and changes sign at α = n-2.
Necessary for the quadratic form of the energy to control the support dimension; referenced in the abstract's discussion of the sign change and absence of dimensionality loss.

pith-pipeline@v0.9.0 · 5734 in / 1513 out tokens · 36984 ms · 2026-05-25T12:26:15.500879+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

[1]

Balagu´ e, J

D. Balagu´ e, J. A. Carrillo, T. Laurent, and G. Raoul. Dimensionality of Local Minimizers of the Interaction Energy. Arch. Ration. Mech. Anal. , 209:1055–1088, 2013

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Burger, B

M. Burger, B. D¨ uring, L. M. Kreusser, P. A. Markowich, an d C. B. Sch¨ onlieb. Pattern formation of a nonlocal, anisotropic interaction model. Math. Models Methods Appl. Sci. , 28:409–451, 2018

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J. A. Ca˜ nizo, J. A. Carrillo, and F. S. Patacchini. Exist ence of compactly supported global minimisers for the interaction energy. Arch. Ration. Mech. Anal. , 217:1197–1217, 2015

work page 2015
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J. A. Carrillo, D. Castorina, and B. Volzone. Ground stat es for diﬀusion dominated free energies with logarithmic interaction. SIAM J. Math. Anal. , 47:1–25, 2015

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J. A. Carrillo, Y.-P. Choi, M. Hauray, and S. Salem. Mean- ﬁeld limit for collective behavior models with sharp sensitivity regions. J. Eur. Math. Soc. (JEMS) , 21:121–161, 2019

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J. A. Carrillo, M. G. Delgadino, and A. Mellet. Regularit y of Local Minimizers of the Interaction Energy Via Obstacle Problems. Comm. Math. Phys. , 343:747–781, 2016

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J. A. Carrillo, M. G. Delgadino, and F. S. Patacchini. Exi stence of ground states for aggregation-diﬀusion equations. Anal. Appl. (Singap.) , 17:393–423, 2019. THE SPHEROID LA W 23

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J. A. Carrillo, S. Hittmeir, B. Volzone, and Y. Yao. Nonli near aggregation-diﬀusion equations: radial symmetry and long time asymptotics. To appear in Invent. Math

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J. A. Carrillo and Y. Huang. Explicit equilibrium soluti ons for the aggregation equation with power-law potentials. Kin. Rel. Mod. , 10:171–192, 2017

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J. A. Carrillo, J. Mateu, M. G. Mora, L. Rondi, L. Scardia , and J. Verdera. The ellipse law: Kirchhoﬀ meets dislocations. To appear in Comm. Math. Phys

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D. Chafa ¨ ı, N. Gozlan, and P.-A. Zitt. First-order glob al asymptotics for conﬁned particles with singular pair repulsion. Ann. Appl. Probab. , 24:2371–2413, 2014

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G. Di Fratta. The Newtonian potential and the demagneti zing factors of the general ellipsoid. Proc. R. Soc. A , 472: 20160197, 2016

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R. C. Fetecau, Y. Huang, and T. Kolokolnikov. Swarm dyna mics and equilibria for a nonlocal aggregation model. Nonlinearity, 24:2681–2716, 2011

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T. Hmidi, J. Mateu, and J. Verdera. On rotating doubly co nnected vortices. J. Diﬀerential Equations , 258:1395–1429, 2015

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O. D. Kellogg. Foundations of Potential Theory . Springer-Verlag, Berlin, 1967

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H. Kunz and C. K. Hemelrijk. Simulations of the social or ganization of large schools of ﬁsh whose per- ception is obstructed. Appl. Anim. Behav. Sci. , 138:142–151, 2012

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M.G. Mora, M. Peletier, and L. Scardia. Convergence of i nteraction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane conﬁnement. SIAM J. Math. Anal. , 49:4149–4205, 2017

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M.G. Mora, L. Rondi, and L. Scardia. The equilibrium mea sure for a nonlocal dislocation energy. Comm. Pure Appl. Math. , 72:136–158, 2019

work page 2019
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E. B. Saﬀ and V. Totik. Logarithmic potentials with external ﬁelds . Springer-Verlag, Berlin, 1997

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Scagliotti

A. Scagliotti. Nonlocal interaction problems in dislocation theory. Tesi di Laurea Magistrale in Matematica, Universit` a di Pavia, 2018

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Seidl and W

R. Seidl and W. Kaiser. Visual ﬁeld size, binocular doma in and the ommatidial array of the compound eyes in worker honey bees. J. Comp. Physiol. A , 143:17–26, 1981

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Simione, D

R. Simione, D. Slepˇ cev, and I. Topaloglu. Existence of ground states of nonlocal-interaction energies. J. Stat. Phys. , 159:972–986, 2015

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E. M. Stein. Singular integrals and diﬀerentiability properties of func tions. Princeton University Press, Princeton, 1970

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E. Wigner. Characteristic vectors of bordered matrice s with inﬁnite dimensions. Ann. Math. , 62:548–564, 1955. (J.A. Carrillo) Department of Mathematics, Imperial College London, Unite d Kingdom E-mail address : carrillo@imperial.ac.uk (J. Mateu) Department de Matem `atiques, Universitat Aut `onoma de Barcelona, Catalonia E-mail address : mateu@mat.uab.c...

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[1] [1]

Balagu´ e, J

D. Balagu´ e, J. A. Carrillo, T. Laurent, and G. Raoul. Dimensionality of Local Minimizers of the Interaction Energy. Arch. Ration. Mech. Anal. , 209:1055–1088, 2013

work page 2013

[2] [2]

Burger, B

M. Burger, B. D¨ uring, L. M. Kreusser, P. A. Markowich, an d C. B. Sch¨ onlieb. Pattern formation of a nonlocal, anisotropic interaction model. Math. Models Methods Appl. Sci. , 28:409–451, 2018

work page 2018

[3] [3]

J. A. Ca˜ nizo, J. A. Carrillo, and F. S. Patacchini. Exist ence of compactly supported global minimisers for the interaction energy. Arch. Ration. Mech. Anal. , 217:1197–1217, 2015

work page 2015

[4] [4]

J. A. Carrillo, D. Castorina, and B. Volzone. Ground stat es for diﬀusion dominated free energies with logarithmic interaction. SIAM J. Math. Anal. , 47:1–25, 2015

work page 2015

[5] [5]

J. A. Carrillo, Y.-P. Choi, M. Hauray, and S. Salem. Mean- ﬁeld limit for collective behavior models with sharp sensitivity regions. J. Eur. Math. Soc. (JEMS) , 21:121–161, 2019

work page 2019

[6] [6]

J. A. Carrillo, M. G. Delgadino, and A. Mellet. Regularit y of Local Minimizers of the Interaction Energy Via Obstacle Problems. Comm. Math. Phys. , 343:747–781, 2016

work page 2016

[7] [7]

J. A. Carrillo, M. G. Delgadino, and F. S. Patacchini. Exi stence of ground states for aggregation-diﬀusion equations. Anal. Appl. (Singap.) , 17:393–423, 2019. THE SPHEROID LA W 23

work page 2019

[8] [8]

J. A. Carrillo, S. Hittmeir, B. Volzone, and Y. Yao. Nonli near aggregation-diﬀusion equations: radial symmetry and long time asymptotics. To appear in Invent. Math

work page

[9] [9]

J. A. Carrillo and Y. Huang. Explicit equilibrium soluti ons for the aggregation equation with power-law potentials. Kin. Rel. Mod. , 10:171–192, 2017

work page 2017

[10] [10]

J. A. Carrillo, J. Mateu, M. G. Mora, L. Rondi, L. Scardia , and J. Verdera. The ellipse law: Kirchhoﬀ meets dislocations. To appear in Comm. Math. Phys

work page

[11] [11]

Chafa ¨ ı, N

D. Chafa ¨ ı, N. Gozlan, and P.-A. Zitt. First-order glob al asymptotics for conﬁned particles with singular pair repulsion. Ann. Appl. Probab. , 24:2371–2413, 2014

work page 2014

[12] [12]

Dal Maso

G. Dal Maso. An introduction to Γ-convergence. Birkh¨ auser, Boston, 1993

work page 1993

[13] [13]

Di Fratta

G. Di Fratta. The Newtonian potential and the demagneti zing factors of the general ellipsoid. Proc. R. Soc. A , 472: 20160197, 2016

work page 2016

[14] [14]

R. C. Fetecau, Y. Huang, and T. Kolokolnikov. Swarm dyna mics and equilibria for a nonlocal aggregation model. Nonlinearity, 24:2681–2716, 2011

work page 2011

[15] [15]

G. B. Folland. Introduction to partial diﬀerential equations. Second edition. Princeton University Press, Princeton, 1995

work page 1995

[16] [16]

Frostman

O. Frostman. Potentiel d’´ equilibre et capacit´ e des ensembles avec quelques applications ` a la th´ eorie des fonctions. Meddel. Lunds Univ. Mat. Sem. , 3:1–118, 1935

work page 1935

[17] [17]

Hirth and J

J.P. Hirth and J. Lothe. Theory of dislocations. John Wiley & Sons, New York, 1982

work page 1982

[18] [18]

Hmidi, J

T. Hmidi, J. Mateu, and J. Verdera. On rotating doubly co nnected vortices. J. Diﬀerential Equations , 258:1395–1429, 2015

work page 2015

[19] [19]

O. D. Kellogg. Foundations of Potential Theory . Springer-Verlag, Berlin, 1967

work page 1967

[20] [20]

Kunz and C

H. Kunz and C. K. Hemelrijk. Simulations of the social or ganization of large schools of ﬁsh whose per- ception is obstructed. Appl. Anim. Behav. Sci. , 138:142–151, 2012

work page 2012

[21] [21]

An Energy Estimate for Dislocation Configurations and the Emergence of Cosserat-Type Structures in Metal Plasticity

G. Lauteri and S. Luckhaus. An energy estimate for dislo cation conﬁgurations and the emergence of Cosserat-type structures in metal plasticity. Preprint arXiv:1608.06155 , 2016

work page internal anchor Pith review Pith/arXiv arXiv 2016

[22] [22]

M.G. Mora, M. Peletier, and L. Scardia. Convergence of i nteraction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane conﬁnement. SIAM J. Math. Anal. , 49:4149–4205, 2017

work page 2017

[23] [23]

M.G. Mora, L. Rondi, and L. Scardia. The equilibrium mea sure for a nonlocal dislocation energy. Comm. Pure Appl. Math. , 72:136–158, 2019

work page 2019

[24] [24]

E. B. Saﬀ and V. Totik. Logarithmic potentials with external ﬁelds . Springer-Verlag, Berlin, 1997

work page 1997

[25] [25]

Scagliotti

A. Scagliotti. Nonlocal interaction problems in dislocation theory. Tesi di Laurea Magistrale in Matematica, Universit` a di Pavia, 2018

work page 2018

[26] [26]

Seidl and W

R. Seidl and W. Kaiser. Visual ﬁeld size, binocular doma in and the ommatidial array of the compound eyes in worker honey bees. J. Comp. Physiol. A , 143:17–26, 1981

work page 1981

[27] [27]

Simione, D

R. Simione, D. Slepˇ cev, and I. Topaloglu. Existence of ground states of nonlocal-interaction energies. J. Stat. Phys. , 159:972–986, 2015

work page 2015

[28] [28]

E. M. Stein. Singular integrals and diﬀerentiability properties of func tions. Princeton University Press, Princeton, 1970

work page 1970

[29] [29]

E. Wigner. Characteristic vectors of bordered matrice s with inﬁnite dimensions. Ann. Math. , 62:548–564, 1955. (J.A. Carrillo) Department of Mathematics, Imperial College London, Unite d Kingdom E-mail address : carrillo@imperial.ac.uk (J. Mateu) Department de Matem `atiques, Universitat Aut `onoma de Barcelona, Catalonia E-mail address : mateu@mat.uab.c...

work page 1955