Free energy minimizers with radial densities: classification and quantitative stability

Lauro Silini; Shrey Aryan

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Free energy minimizers with radial densities: classification and quantitative stability

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Pith's one-line read This paper shows that, for monotone increasing strictly admissible radial weights, centered spheres uniquely minimize the free energy at every volume, and the energy gap controls the squared weighted distance to the ball.

desk verdict Counterexamples genuinely answer Q1 in the negative; the monotone classification is a serious Chambers adaptation; the stability theorem has a repairable but real gap in Proposition 4.2/Remark 4.3. read the letter →

arxiv 2412.03997 v2 pith:VTDGOGYJ submitted 2024-12-05 math.AP

Shrey Aryan , Lauro Silini This is my paper · ORCID

classification math.AP MSC 49Q2049Q10

keywords weightedisoperimetricproblemfreeenergyminimizersradialdensitysphericalsymmetrizationquantitativestabilityprofilemonotoneweightsnearlysets

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 studies minimizers of a free-energy functional $E(F)=P_f(F)+G_f(F)$ on $\mathbb{R}^n$, where $f=e^\psi$ weights the perimeter and $g$ weights the bulk potential, with both $\psi$ and $g$ radially symmetric. The central claim is a complete answer to the question of when centered spheres are the unique global minimizers under a fixed weighted volume. Local stability of the spheres is equivalent to $\psi''+g'\ge 0$, but the paper shows this condition alone is not sufficient in dimension $n\ge2$, constructing two counterexamples. Adding the assumption that both $\psi$ and $g$ are monotone increasing (with strict admissibility, meaning $\psi''+g'>0$ for $r>0$, together with the standing growth and regularity hypotheses) forces the centered spheres to be the unique minimizers for every volume. From this classification the paper derives an explicit isoperimetric profile and a sharp quadratic stability inequality, in which the energy excess of a same-volume competitor controls the squared $f$-weighted symmetric difference to the ball.

What carries the argument

The classification argument runs through spherical symmetrization, which replaces any competitor by a spherically symmetric set with no larger energy and the same weighted volume. The boundary of the symmetrized minimizer is described by a plane curve $\gamma$ solving the constant weighted mean curvature equation (3.3), and the proof decomposes $\gamma$ into an upper curve, a lower curve, and a curl curve; a comparison argument shows the lower curve bends faster than the upper one, so it cannot cross the symmetry axis, and any curling contradicts the tangent restriction, forcing $\gamma$ to be a centered circle. A separate calibration argument, using a radial vector field with an explicit multiplier, gives large-volume uniqueness under $\kappa$-uniform admissibility. For stability, the key machinery is a second-order expansion on nearly spherical sets, writing $\partial E=\{Rx(1+u(x))\}$, expanding the energy gap to second order, and controlling it from below by $\|u\|^2_{L^2}$; the local estimate is then connected to global sets through almost-minimizers, uniform boundedness, and a compactness argument.

What would settle it

Evaluate condition (4.3) along the radii $r_k$ produced by Lemma 4.14 for a monotone strictly admissible pair; in $n=2$ with $\psi(r)=\arctan r$ the condition reduces to $\psi'(r_k)>r_k$, which fails for all $r_k>1$, so if the contradiction argument reaches such radii the proof of Theorem 1.8 as written cannot invoke Proposition 4.2.

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Extended reading notes

Core claim

The central discovery is that the stability condition $\psi''+g'\ge 0$, which characterizes local minimality of centered spheres, is not enough to guarantee global optimality once a potential term $g$ is present. In dimension $n\ge2$, the paper constructs $\kappa$-uniformly admissible weights with $\psi$ minimized at the origin for which non-centered balls beat the centered ball at intermediate volumes, so monotonicity of the two weights is needed. Under monotone increasing strictly admissible hypotheses, Theorem 1.5 establishes that centered spheres uniquely solve the volume-constrained problem for every volume; Corollary 1.6 records the resulting isoperimetric profile identity, and Theorem 1.8 states a sharp quadratic stability bound $E(E)-E(B_R)\ge c|E\triangle B_R|_f^2$ for all sets of the same weighted volume.

Load-bearing premise

The stability proof requires that at every radius $r_k$ selected by the compactness argument the local estimate's condition (4.3) holds, meaning $\psi'(r_k)>-\frac{(n-2)(n-1)}{r_k^2(\psi'(r_k)+g(r_k))}+(n-1)r_k$; this domination is not implied by the monotone strictly admissible hypotheses, and for $n=2$, $\psi(r)=\arctan r$ it fails for every $r>1$.

Editorial extensions

If this is right

For any monotone increasing strictly admissible pair $(\psi,g)$, the volume-constrained variational problem is fully solved: every minimizer is a centered ball, so further analysis of the free energy can be restricted to balls.
The explicit isoperimetric profile formula $E'(v)=g(r)+\psi'(r)+(n-1)/r$ with $r=\Phi^{-1}(v)$ makes it possible to detect where the profile is concave or convex and to locate volumes at which centered uniqueness may fail.
The two counterexamples show that numerical or variational methods that rely only on $\psi''+g'\ge0$ to certify global optimality will fail in dimension $n\ge2$; monotonicity checks must be added.
The sharp quadratic stability inequality implies quantitative convergence of any minimizing sequence to the centered ball in the $f$-weighted $L^1$ distance, with the exponent 2 optimal since ellipsoidal deformations saturate the bound.
In the large-volume regime, $\kappa$-uniform admissibility alone guarantees that centered spheres of radius larger than $\sqrt{n+2}/\kappa$ are the unique minimizers.

Reading between the lines

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

A natural extension not pursued in the paper is to lower the regularity requirement on the monotone weights; the comparison structure of the curve argument suggests that a density/approximation argument could preserve the classification for merely $C^2$ monotone strictly admissible weights.
The counterexamples occupy the intermediate-volume regime, which is the natural frontier: small volumes concentrate near minima of $e^\psi$, large volumes are forced to centered balls by calibration, and the failure of global optimality is a genuinely intermediate-volume phenomenon; a one-parameter family of weights could be tested numerically across this transition.
For the stability theorem, the proof would become unconditional if the local estimate of Proposition 4.2 were stated directly under the weaker condition displayed in Remark 4.3 and that condition were shown to hold at the radii produced by the compactness argument; making this substitution is a concrete open step.
The sharpness of the quadratic exponent via ellipsoids suggests that the optimal constant in (1.4) is governed by the lowest eigenvalue of the second variation; computing it for a model potential such as a linear field would give an explicit quantitative bound for droplet shapes in a gravitational field.

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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

The thing to know: this paper answers a natural question — whether ψ′′+g′ ≥ 0 suffices for centered spheres to minimize the free energy P_f + G_f — and the answer is no. The two counterexamples (Propositions 2.16 and 2.17) are genuinely new and reasonably clean: the stability condition holds, ψ has its minimum at the origin, and a displaced ball still beats the centered one at intermediate volumes. That alone makes the paper worth reading for anyone in weighted isoperimetric problems.

The positive results are also real. Theorem 1.5, the classification under monotone increasing ψ and g, is a careful adaptation of Chambers' log-convex density proof to a functional with a potential. Much of the machinery is imported, from Chambers and from Boyer–Brown–Chambers–Loving–Tammem, but the adaptation is the contribution, and it appears to hold together. The one-dimensional characterization (Theorem 1.2) is new and sharp, and the large-volume result (Theorem 1.3) is a solid minor extension of the Kolesnikov–Zhdanov calibration.

The soft spot is Theorem 1.8, the quantitative stability estimate. Proposition 4.2 states condition (4.3), and Remark 4.3 claims it is automatic under the stated hypotheses. That claim is false: with n = 2, ψ(r) = arctan(r), g(r) = r + C, the hypotheses of Theorem 1.5 hold but (4.3) fails for large R. So the application of Proposition 4.2 at the radii r_k in the contradiction argument of Section 4.2 is not justified as written. The stress-test note is right that this is repairable: the proof only needs the weaker condition stated inside Remark 4.3, which does follow from monotone increasing weights. So Theorem 1.8 likely survives with a corrected statement and proof. There are also some inconsistent displayed formulas in Proposition 2.16 and Lemma 3.9 — minor, but they should be fixed.

The citation pattern is fine. Heavy reliance on [11] and [15] is legitimate because those are published proofs; the counterexamples and the classification are not circular.

Who this is for: people working on weighted isoperimetric problems, free boundary problems with external potentials, and quantitative stability. It deserves a serious referee. I would send it out, asking the authors to fix the (4.3)/Remark 4.3 issue and clean up the displayed formulas. The main classification and the counterexamples look solid.

Recommendation: engage with it — referee it if you get the chance, and read the counterexamples either way.

Referee Report

2 major / 6 minor

Summary. The paper studies the free-energy isoperimetric problem in R^n with perimeter density f=e^ψ and external potential g, both radial, under a volume constraint with respect to f. After establishing existence, boundedness, and regularity of minimizers, it computes the first and second variations and shows that centered spheres are stationary and stable exactly when ψ''+g'≥0. The main results are: (i) a complete one-dimensional classification; (ii) two counterexamples in n≥2 showing that ψ''+g'>0 plus a minimum of ψ at the origin does not suffice for global optimality of centered spheres; (iii) a global uniqueness theorem for centered spheres when ψ and g are both monotone increasing and strictly admissible (Theorem 1.5); and (iv) a sharp quadratic stability inequality in that monotone class (Theorem 1.8). The proof of (iii) is an adaptation of Chambers' proof of the log-convex density conjecture, and the proof of (iv) follows the Fuglede-type scheme of Fusco–La Manna.

Significance. If the proofs are completed, the paper gives a fairly complete answer to the question of when the stability condition ψ''+g'≥0 is also sufficient for global optimality of centered spheres, and it provides a quantitative rigidity statement in the monotone regime. The two counterexamples are instructive and show that the one-dimensional behavior does not persist in higher dimensions. The main classification is a substantial adaptation of Chambers' argument to the case with a potential, and the quantitative stability result is a natural and useful strengthening. The paper also contributes elementary existence and regularity results for this free-energy functional. The proofs are mostly self-contained and the statements are precise, with the qualification that the quantitative stability half currently contains two repairable gaps, detailed below. The counterexamples and the classification argument are, in my reading, sound.

major comments (2)

[Section 4.1 (Proposition 4.2 and Remark 4.3)] Condition (4.3) in Proposition 4.2 is not implied by the hypotheses of Theorem 1.8, and the assertion in Remark 4.3 that (4.3) is automatically satisfied for monotone increasing strictly admissible weights is false. For example, take n=2, ψ(r)=arctan(r), and g(r)=r+C with C>0. Then ψ is increasing, g is increasing, and ψ''(r)+g'(r)=1-2r/(1+r^2)^2>0 for all r>0, so the weights satisfy the assumptions of Theorem 1.8. However, for n=2 the right-hand side of (4.3) reduces to R, while ψ'(R)=1/(1+R^2)<R for all sufficiently large R, so (4.3) fails. Since the proof of Theorem 1.8 in Section 4.2 applies Proposition 4.2 at the radii r_k produced by Lemma 4.14, with r_k converging to the arbitrary radius R>0, the proof of Theorem 1.8 is not justified as written. This is repairable: the proof of Proposition 4.2 only requires the weaker condition stated in Remark 4.3, namely (R^2ψ'(R)+R(n-1))(ψ'(R)+g(R))+R(n-1)ψ'(R)+R^2(ψ''(R)+g'(R))+(n-1)(n-2)>0, which is automatically satisfied when ψ'≥0, g≥0 and ψ''+g'>0. The proposition should be restated with that weaker hypothesis and the remark amended.
[Section 4.2 (Theorem 1.8 proof, Lemma 4.7)] The proof of Theorem 1.8 uses Lemma 4.7 in the case of large asymmetric difference, but Lemma 4.7 requires the hypothesis ψ(r)>ψ(0). This is not a consequence of the assumptions 'ψ,g monotone increasing strictly admissible' if 'monotone increasing' is understood in the standard non-decreasing sense, which the paper's own usage in Proposition 1.4 and Proposition 2.17 suggests. For instance, ψ≡0 and g(r)=r are monotone increasing and strictly admissible, yet ψ(r)=ψ(0) for every r. The contradiction argument of Lemma 4.7 would still need a separate justification in this degenerate case; as written, the application of Lemma 4.7 in the proof of Theorem 1.8 is not covered by its hypotheses. This is a further gap in the stability theorem, though it is also local and repairable, either by treating the constant-ψ case separately (e.g., via [43] when ψ≡0) or by a limiting argument from strictly increasing perturbations.

minor comments (6)

[Section 2.1 (Proposition 2.2 and Remark 2.3)] The statement of Proposition 2.2 uses condition (2.2), but Remark 2.3 and the proof refer to 'condition (2.3)' and 'thanks to condition (2.3)'; these references should be to (2.2).
[Section 2.4 (Proposition 2.16)] The estimate e^{M/n}-e^{M/(2n)} ≥ M/n used near the end of the proof is not correct as written; the correct lower bound is e^{M/n}-e^{M/(2n)} ≥ M/(2n). The conclusion remains valid because the second term on the right-hand side of (2.25) still has a factor ε^{n-1} and can be made small, but the displayed constant in the perimeter estimate should be adjusted.
[Section 3.1 (Proposition 3.2)] The last display of Proposition 3.2 reads Pf(F⋆ξ) ≤ Pf(Fξ); the right-hand side should be Pf(F), not Pf(Fξ).
[Section 4.1 (Lemma 4.14)] In the statement of Lemma 4.14, the phrase 'for each i ∈ N' is unused and appears to be a leftover; the graph u_k should be indexed by k, and the regularity statement 'C^{1,α} for all α<1/2' should be stated consistently.
[Section 2.3 (Theorem 1.3 proof)] In the lower bound for h, the expression ℓ′ + ℓ r + (n-1)/r should read ℓ′ + ℓ(r + (n-1)/r); the factor ℓ is missing in front of (n-1)/r.
[Section 3.2 (Lemma 3.9(2))] The displayed computation of ˜H''_1(0) contains unreadable OCR artifacts (the strings '/bracehtipupleft/bracehtipdownright/...'); these should be cleaned up in the LaTeX source.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from symmetrization, calibration, and ODE comparison with external references, not from the authors' own conclusions.

full rationale

I examined the derivation chain for Theorems 1.5 and 1.8. The classification proof in Section 3 proceeds by spherical symmetrization (Proposition 3.2), reduction to a generating curve satisfying the weighted mean curvature ODE (Lemma 3.4), and a geometric upper/lower/curl curve argument adapted from Chambers [15]; the cited geometric lemmas are external, prior work by other authors, and not by the present authors. The counterexamples in Propositions 2.16 and 2.17 are explicit constructions with free parameters (M, epsilon, L, h, delta) chosen through inequalities, not fitted to the conclusion. The stability result in Section 4 uses a self-contained second-order expansion for nearly spherical sets in Proposition 4.2 and the Fusco--La Manna compactness scheme from [33]; the authors' own prior works [64,65] appear only in the introduction as context and are not load-bearing. The weak point noted by the reader -- that hypothesis (4.3) in Proposition 4.2 is not implied by Theorem 1.8's hypotheses -- is a correctness gap in a displayed sufficient condition, not a circularity: the proof only needs the weaker condition recorded in Remark 4.3, and that weaker condition does not assume the theorem's conclusion. No equation is defined in terms of the target result, no fitted parameter is renamed as a prediction, and no load-bearing step reduces to a self-citation. Therefore the derivation is self-contained with respect to circularity.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central theorems introduce no fitted parameters. The free parameters listed belong to the counterexample constructions, where constants are chosen by hand to satisfy inequalities; they do not feed back into the main classification. The axioms are the stated symmetry, monotonicity, and admissibility assumptions plus standard geometric measure theory and imported results from [15], [11], and [45]. The only ad hoc axiom is condition (4.3) in Proposition 4.2, which the paper treats as automatically satisfied but is not.

free parameters (6)

M (plateau height in Prop 2.16) = arbitrary large M > 0
Sets the scale of the energy advantage for the off-center competitor B'; chosen by hand to make the perimeter gap dominate.
epsilon (transition width in Prop 2.16) = small, satisfying (2.23)
Controls the slope of psi near 0 and the error estimate in (2.25); chosen after the volume v is fixed.
h = L' - L = large, satisfying (2.24)
Makes the potential gap contribution (M/h)v negligible compared with the perimeter gain.
C (additive constant in g) = large
Ensures g > 0 in Proposition 2.16.
delta (linear perturbation added to g) = small
Forces kappa-uniform admissibility without reversing the energy inequality.
g(0) in Prop 2.17 = large
Keeps g positive while prescribing g(r) = g(0) - psi'(r) on the intermediate interval.

assumptions (7)

domain assumption Radial symmetry of psi and g with respect to the origin.
The problem (∗) is defined only for symmetric weights; the paper does not derive symmetry from a variational principle.
domain assumption Admissibility: psi in C^2, g in C^1, psi'(0)=0, and psi''+g' >= 0 for all r >= 0.
Definition 1.1; this is the local stability condition, which the paper shows is not sufficient in general.
domain assumption Monotone increasing psi and g for Theorems 1.5 and 1.8.
The positive classification and stability results are conditional on this added monotonicity.
standard math Standard geometric measure theory: BV compactness, coarea formula, regularity theory for omega-minimizers.
Used in Theorems 2.1, 2.4, 2.5 and in the stability section; cited to [47,48] and [33].
standard math Kolesnikov-Zhdanov calibration lemma.
Used in the proof of Theorem 1.3 to show centered balls minimize for large volumes; imported from [45].
standard math Chambers' log-convex density theorem and the auxiliary geometric lemmas of Boyer-Brown-Chambers-Loving-Tammen.
The curve analysis in Section 3 is an adaptation of [15] and [11]; the paper relies on several lemmas from these references without full proofs.
ad hoc to paper Condition (4.3) in Proposition 4.2 is available at the radii used in the proof of Theorem 1.8.
This condition is introduced for the local stability estimate; it is not among the hypotheses of Theorem 1.8 and Remark 4.3's claim that it follows automatically is false.

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Pith. "Pith review of Free energy minimizers with radial densities: classification and quantitative stability." pith.science (2026). https://pith.science/paper/VTDGOGYJ

@misc{pith2026241203997,
  author       = {Pith},
  title        = {Pith review of: Free energy minimizers with radial densities: classification and quantitative stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTDGOGYJ}},
  note         = {Machine review of arXiv:2412.03997}
}

abstract

We study the isoperimetric problem with a potential energy $g$ in $\mathbb{R}^n$ weighted by a radial density $f$ and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast to the classical isoperimetric case $g = 0$, the condition $\ln(f)'' + g' \geq 0$ does not generally guarantee the global optimality of centered spheres. However, we demonstrate that centered spheres are globally optimal when both $f$ and $g$ are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality.

Figures

Figures reproduced from arXiv: 2412.03997 by the authors.

**Figure 1.** First construction of ψ and g. so that ψ ′′ + g ′ ≥ 0 in [0, L′′], proving that ψ, g are admissible weights. Moreover, taking C > 0 big enough, we can enforce g > 0 everywhere. Call Bε the ball centered at the origin with radius ε > 0. Notice that with the choice of the constants we did at the beginning, the ball B centered at the origin with weighted volume v has radius τ > 0 strictly between ε and L, since |Bε|f <… view at source ↗

**Figure 2.** Construction of ψ. Proof. It is enough to construct ψ and g to be admissible weights, since arguing exactly as the end of the proof of Proposition 2.16, it is sufficient to add a small enough linear term to g in order ensure κ-uniformly admissibility. We construct ψ as in the [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

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