A family of explicit minimizers for interaction energies

Ruiwen Shu

A family of explicit minimizers for interaction energies

T0 review · 0 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proves that for odd dimensions with potential exponents $(a,b)=(3,2-d)$ and even dimensions with $(a,b)=(3,1-d)$, the unique minimizer of the power-law interaction energy has an explicit formula, obtained by solving a local PDE…

desk verdict Solid explicit-minimizer result: the construction is sound, the new parameter families are genuinely new, and the reliance on cited uniqueness theory is appropriate. read the letter →

arxiv 2501.14666 v1 pith:L3ZHADAX submitted 2025-01-24 math.AP

Ruiwen Shu This is my paper · ORCID

classification math.AP MSC 35A1545E10

keywords interactionenergypower-lawpotentialexplicitminimizerEuler-LagrangeconditiondimensionreductionradialsymmetryuniquenessLaplacian

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives explicit formulas for the unique minimizer (up to translation) of the interaction energy with power-law potential $W(x)=|x|^a/a - |x|^b/b$ in $d$ dimensions, for odd $d$ with $(a,b)=(3,2-d)$ and even $d$ with $(a,b)=(3,1-d)$. These are new families in the catalogue of explicit minimizers, where previously known examples included quadratic attractive cases, spherical shells, and one-dimensional power laws. In odd dimensions the key step is to apply the Laplacian repeatedly to the Euler-Lagrange condition $W*\rho=C_0$ on the support, turning a nonlocal convolution condition into a local linear PDE whose radial power-series solutions are explicit; the support radius is then selected by a determinantal condition. In even dimensions the minimizer is the projection and rescaling of the minimizer in one higher dimension, which the paper proves by a new dimension-reduction lemma. In dimensions 1, 2, and 3 the formulas reduce to elementary functions, such as $\rho_1(x)=C\cosh(\sqrt{2}\,x)$ on its support.

What carries the argument

The central object is the Laplacian ladder: for the chosen exponent pair, applying $\Delta$ successively to the Euler-Lagrange equation $W*\rho=C_0$ on the support produces the constant-coefficient linear PDE $\Delta^{(d+1)/2}\rho=A_*\rho$, which is local and can be solved in radial coordinates by power series. Solving it yields the basis functions $u_{2k}$ and the coefficient matrix $M(R)$, whose determinant fixes the support radius; the same condition also picks the correct coefficients in the nullspace. For even dimensions, the projection operator $P$ (integration over the extra coordinate) and the averaging operator $Q$ are the central mechanisms: Lemma 4.1 shows that $W*F=C$ in dimension $d+1$ is equivalent to $\tilde{W}*P[F]=C$ in dimension $d$, and Lemma 5.1 shows that power-law potentials are mapped by $Q$ to power-law potentials up to constants, up to the scaling factor $\lambda$.

What would settle it

Compute for $d=5$ the entries of $M(R)$ from (3.23) using the recurrence (3.17) truncated at large order, find every positive root of $\det M(R)=0$, and check whether the nullspace vector yields $\rho_{R,c}\ge 0$ on $B(0,R)$ and whether $W*\rho_{R,c}$ is constant on the support. A high-resolution numerical minimization of the energy in $d=2$ compared with the explicit formula (1.10) would also settle the claim: any lower-energy competitor would contradict uniqueness.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: for odd $d$, the unique minimizer of $E$ is the normalized radial function $\rho_{R,c}$ supported on $B(0,R)$, built from the power series (3.19)-(3.20), where $R$ is the unique positive solution of $\det M(R)=0$ whose associated coefficient vector lies in the one-dimensional nullspace and makes the density nonnegative. The matrix $M(R)$ encodes the repeated-Laplacian conditions evaluated at the origin, and its entries are explicit power series in $R$. For even $d$, Corollary 5.3 states that the minimizer is $\tilde{\rho}(x_1,\ldots,x_d)=\lambda^d \int_{\mathbb R} \rho(\lambda x_1,\ldots,\lambda x_d,x_{d+1})\,dx_{d+1}$ with an explicit scaling $\lambda$, so the even-dimensional minimizer is a projected and rescaled copy of the odd-dimensional one. The paper also writes the minimizers in dimensions 1, 2, and 3 in elementary closed form, with the support radii fixed by scalar equations such as $\sqrt{2}R=\coth(\sqrt{2}R)$.

Load-bearing premise

The load-bearing premise is that the imported uniqueness theory applies to these exponent pairs, so any ball-supported radial density whose convolution with the potential is constant on its support must be the unique global minimizer; if that characterization failed, the constructed candidates would be only steady states.

Editorial extensions

If this is right

In odd dimensions, the unique minimizer is real-analytic on its support, so its profile can be evaluated to arbitrary precision by truncating the defining power series.
In dimensions 1, 2, and 3, the elementary closed forms allow direct computation of the support radius and of the minimal interaction energy without numerical integration of the convolution.
In even dimensions, the minimizer is exactly a projection and rescaling of the minimizer in dimension $d+1$, so explicit minimizers propagate from one dimension to the next within this family.
Because these potentials satisfy linear interpolation convexity, verifying the Euler-Lagrange condition is sufficient, so the constructed densities are true global minimizers rather than merely steady states.
The same repeated-Laplacian procedure produces radial functions satisfying $W*\rho=C$ on a ball for other integer-parity exponent pairs, although these are only steady states unless nonnegativity and convexity are independently verified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the Laplacian-ladder mechanism should extend to powers $a>3$ whenever the successive derivatives of $|x|^a/a$ stay integrable and eventually become local; testing the same construction for $(a,b)=(5,2-d)$ against numerical minimizers would show whether the method is a general template.
Editorial inference: the dimension-reduction lemma is not tied to power-law potentials, so any new explicit minimizer in odd dimensions could immediately produce an even-dimensional minimizer through projection, provided the averaged potential retains the same structure.
Editorial inference: the explicit densities in dimensions 1, 2, and 3 are natural benchmark solutions for numerical solvers for aggregation equations, since they have known support radii, known energy values, and non-smooth behavior at the boundary that stress numerical methods.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a solid paper, and it delivers more than the abstract promises. The explicit minimizers for (a,b)=(3,2-d) in odd dimensions and (3,1-d) in even dimensions are genuinely new — none of the previously known families listed in the introduction cover these exponents. The formulas for d=1,2,3 are explicit enough to plot, and the general construction is coherent.

What I like: the Laplacian cascade is a nice idea. Taking successive Laplacians of the Euler-Lagrange condition turns the nonlocal equation into a local ODE, which can be solved by power series. The stress-test note checks out: the induction showing that (3.10) at r=0 implies the full Euler-Lagrange equality (3.8) on the ball is valid. The dimension-reduction lemma (Lemma 4.1) is a clean, reusable tool. The paper is also honest about what it imports — the uniqueness and EL-characterization theory from [CS23, CDM16] is standard and the hypotheses are met for these exponent pairs. The citation pattern is fine; the only self-citation, [CS23], is to the exact external result being used as a benchmark.

Soft spots, in proportion: the main one is not a flaw. For general odd d, the "explicit" formula requires solving a transcendental equation and checking nonnegativity, and Remark 3.2 admits that uniqueness of the correct R in high dimensions is not proved. So in higher dimensions the result is semi-constructive rather than fully explicit. For d=1 and d=3 it is fully explicit, which is where the paper's examples land. The paper also leans on Proposition 2.1 without reproving it; if that theory had a gap, the constructed candidates would only be steady states rather than minimizers. But that theory is published and standard, so I do not count this against the paper.

Minor caveat: the computations are long and not machine-checked, so there is room for algebra slips, but I traced the key induction and it holds.

Who this is for: people working on aggregation equations, interaction energies, and explicit equilibria. It will be cited in that community. It deserves a serious referee; my recommendation is to send it out. I would accept it after minor revisions, mainly to sharpen Remark 3.2 and to add a sentence on how the transcendental equation is solved numerically in practice.

Referee Report

0 major / 4 minor

Summary. This paper studies minimizers of the interaction energy E(ρ) = (1/2)∫∫ W(x−y) dρ(y) dρ(x) with W(x)=|x|^a/a − |x|^b/b on R^d. It obtains explicit global minimizers in two new parameter families: odd d with (a,b)=(3,2−d) and even d with (a,b)=(3,1−d). For odd d, the Euler−Lagrange identity W∗ρ=C0 on the support is differentiated (d+1)/2 times; the singular repulsive part is annihilated by successive Laplacians, yielding the local PDE Δ^{(d+1)/2}ρ=A∗ρ. A radial power-series ansatz then reduces the problem to a linear system M(R)c=0 for the power-series coefficients and a scalar equation det M(R)=0 for the support radius. Theorem 3.1 states that the unique positive root R whose nullspace produces a nonnegative density gives the unique minimizer up to translation. For even d, a new projection lemma (Lemma 4.1) shows that steady states in R^{d+1} project to steady states in R^d for a modified potential; after a rescaling this yields the minimizer in even dimensions (Theorem 5.2 and Corollary 5.3). Explicit elementary formulas are supplied for d=1,2,3.

Significance. If correct, these are new explicit global minimizers for power-law interaction energies, extending the previously known polynomial densities, spherical shells, and one-dimensional examples. The construction is genuinely explicit: all inputs are power-series coefficients and the solution of the scalar equation det M(R)=0, and no fitted quantity is fed back into the verification. The proof combines two solid ingredients: the imported LIC/existence/uniqueness theory in Proposition 2.1 and a new dimension-reduction lemma. The paper also honestly records its limitations, notably in Remark 3.2 and in the closing paragraph of Section 3. I checked the key distributional identities and the projection argument; the mathematics appears sound, and the paper is suitable for publication after minor clarifications.

minor comments (4)

[§3.2, proof of Theorem 3.1] The induction step from M(R)c=0 to the full system (3.10) on B(0,R) is stated in one sentence: “one can use induction to show that (3.10) for these k are satisfied.” Since this is the central mechanism of the construction, please spell out the H_k argument: define H_k = A_{2k}|x|^{1−2k}∗ρ − |S^{d−1}|Δ^kρ, verify ΔH_k = H_{k+1} for k=0,…,m−1 and ΔH_m=0 from (3.12), and then use H_k(0)=0 together with radial harmonicity to conclude H_k≡0. Adding these three lines will make the proof self-contained and easier to verify.
[§4, Lemma 4.1] In the proof that Q[U]=0 implies U=0 on B^d(0,R), the sentence about preimages of zero-measure sets under dist(x∨v, x) should be justified more carefully: for fixed x with |x|>0 the map v↦dist(x∨v,x) is smooth with nonvanishing Jacobian away from the two critical points, so the coarea formula gives the required null-preimage property; the current phrasing is a little too casual for a measure-theoretic argument.
[References] The entry [CCP15] contains a duplicated and garbled title (“Existence of compactly supported global minimisers for the interaction existence of compactly supported global minimisers for the interaction energy”); please restore the correct title.
[§3.3, equations (3.28)–(3.30)] The notation βR is used without a separator in equations such as (3.28) and (3.30); although the meaning is clear, typesetting it as βR with a thin space would improve readability.

Circularity Check

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No circularity: the explicit minimizers are constructed from the Euler-Lagrange hierarchy and verified with an external LIC uniqueness theory; the author-overlap citation to [CS23] is real evidence, not a fitted input.

full rationale

The derivation chain is not circular. In Theorem 3.1, the candidate ρ_{R,c} is generated by solving the Laplacian hierarchy; the condition M(R)c=0 is only a set of pointwise conditions at r=0, and the proof explicitly uses an induction (via the radial Laplacian identities) to lift these to the full Euler-Lagrange identity (3.8) on the ball. Thus the candidate is not assumed to satisfy the target by construction. Sufficiency is delegated to Proposition 2.1, whose hypotheses are met for (a,b)=(3,2-d) and (3,1-d). The uniqueness and Euler-Lagrange characterization parts of Proposition 2.1 are cited to [CS23, Theorem 2.4] and [BCLR13a, Theorem 4]; [CS23] is co-authored by the present author, but it is a parameter-free theorem about the whole LIC class of power-law potentials, does not assume the explicit family constructed here, and is externally falsifiable, so under the stated rules it counts as real evidence rather than circularity. The even-dimensional result in Corollary 5.3 is derived from the new projection Lemma 4.1 and a proven scaling identity, not from a fitted parameter renamed as a prediction. No equation in the paper reduces to its own input, and no fitted quantity is fed back into the derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameter is fitted to data or chosen ad hoc: the support radius R and coefficients c_{2k} are determined as solutions of the determinant equation and nullspace system, and the constants R1, R3, lambda are computed from explicit formulas. The proof rests on the imported LIC, uniqueness, and regularity theory summarized in Proposition 2.1, which acts as the external benchmark for verifying the candidates.

assumptions (5)

domain assumption The linear interpolation convexity (LIC) property holds for -d<b<=2<=a<=4, b<a except (a,b)=(4,2), implying uniqueness of the minimizer up to translation.
Invoked in Proposition 2.1 and throughout; the paper relies on [Lop19, CS23, Fra22] rather than reproving it.
domain assumption The Euler-Lagrange condition W*rho=C0 on supp rho, W*rho>=C0 elsewhere, is necessary and sufficient for global minimality under LIC.
Used to identify constructed steady states with true minimizers; cited to [BCLR13a] and [CS23] in Proposition 2.1.
domain assumption For -d<b<=2-d, the minimizer is compactly supported on a ball, has the stated regularity, and is the only radial function in its regularity class with ball support satisfying W*rho=C0 on the support.
This is the bridge from constructed candidates to minimizers in Theorem 3.1 and Theorem 5.2; imported from [CDM16] and the last part of Proposition 2.1.
domain assumption Existence of compactly supported global minimizers for the power-law interaction energies.
Needed so that the minimizer is a well-defined probability measure; cited to [CCP15]/[SST15].
standard math Standard distributional identities for Laplacians of radial functions, including Delta |x|^p = p(p+d-2)|x|^(p-2) away from the origin and Delta |x|^(2-d) proportional to the Dirac mass.
Used in Section 3 to derive the local PDE and in Lemma 5.1 for angular averages.

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Pith. "Pith review of A family of explicit minimizers for interaction energies." pith.science (2026). https://pith.science/paper/L3ZHADAX

@misc{pith2026250114666,
  author       = {Pith},
  title        = {Pith review of: A family of explicit minimizers for interaction energies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3ZHADAX}},
  note         = {Machine review of arXiv:2501.14666}
}

abstract

In this paper we consider the minimizers of the interaction energies with the power-law interaction potentials $W({\bf x}) = \frac{|{\bf x}|^a}{a} - \frac{|{\bf x}|^b}{b}$ in $d$ dimensions. For odd $d$ with $(a,b)=(3,2-d)$ and even $d$ with $(a,b)=(3,1-d)$, we give the explicit formula for the unique energy minimizer up to translation. For the odd dimensions, the key observation is that successive Laplacian of the Euler-Lagrange condition gives a local partial differential equation for the minimizer. For the even dimensions $d$, the minimizer is given as the projection and rescaling of the previously constructed minimizer in dimension $d+1$ via a new lemma on dimension reduction.

Figures

Figures reproduced from arXiv: 2501.14666 by the authors.

**Figure 1.** The minimizers for pd, a, bq being p1, 3, 1q, p2, 3, ´1q and p3, 3, ´1q (left, middle and right, respectively). The blue curves are the minimizers ρ in the radial coordinate. The red curves are the generated potentials W ˚ ρ, subtracted by the constant C0 “ 1 2Erρs. Since the potentials we are considering are LIC, it suffices to verify the Euler-Lagrange condition in order to justify the explicit minimizers. For the… view at source ↗

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Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Balagu \'e , J

D. Balagu \'e , J. A. Carrillo, T. Laurent, and G. Raoul. Dimensionality of local minimizers of the interaction energy. Archive for Rational Mechanics and Analysis , 209(3):1055--1088, 2013

work page 2013

[2] [2]

Balagu \'e , J

D. Balagu \'e , J. A. Carrillo, T. Laurent, and G. Raoul. Nonlocal interactions by repulsive--attractive potentials: Radial ins/stability. Physica D , 260:5--25, 2013

work page 2013

[3] [3]

J. A. Ca \ n izo, J. A. Carrillo, and F. S. Patacchini. Existence of compactly supported global minimisers for the interaction existence of compactly supported global minimisers for the interaction energy. Arch. Ration. Mech. Anal. , 217(3):1197--1217, 2015

work page 2015

[4] [4]

J. A. Carrillo, M. G. Delgadino, and A. Mellet. Regularity of local minimizers of the interaction energy via obstacle problems. Comm. Math. Phys. , 343(3):747--781, 2016

work page 2016

[5] [5]

Choksi, R

R. Choksi, R. C. Fetecau, and I. Topaloglu. On minimizers of interaction functionals with competing attractive and repulsive potentials. Ann. IHP , 32:1283--1305, 2015

work page 2015

[6] [6]

J. A. Carrillo and Y. Huang. Explicit equilibrium solutions for the aggregation equation with power-law potentials. Kinetic and Related Models , 10(1), 2017

work page 2017

[7] [7]

J. A. Carrillo, J. Mateu, M. G. Mora, L. Rondi, L. Scardia, and J. Verdera. The ellipse law: K irchhoff meets dislocations. Commun. Math. Phys. , 373:507--524, 2020

work page 2020

[8] [8]

J. A. Carrillo, J. Mateu, M. G. Mora, L. Rondi, L. Scardia, and J. Verdera. The equilibrium measure for an anisotropic nonlocal energy. Calc. Var. Partial Differential Equations , 60:109, 2021

work page 2021