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arxiv: 2605.01547 · v1 · submitted 2026-05-02 · 🧮 math.AP

Rigidity for the P\'olya-Szeg\"o inequality under circular rearrangement

F. Cagnetti , G. Domazakis , M. Perugini , F. Seuffert This is my paper

Pith reviewed 2026-05-09 17:55 UTC · model grok-4.3

classification 🧮 math.AP
keywords Pólya-Szegő inequalitycircular rearrangementrigiditysymmetric extremalsrearrangement inequalitiesDirichlet integralvariational problems
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The pith

A Pólya-Szegő inequality holds for circular rearrangements with rigidity under sufficient conditions.

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

The paper proves a version of the Pólya-Szegő inequality that compares the gradient integral of a function to the same integral for its circular rearrangement. This holds under general assumptions on the functions and domains. The authors further supply sufficient conditions under which every function attaining equality must itself be symmetric. Such results matter because rearrangement inequalities are standard tools for bounding energies and characterizing solutions in variational problems and elliptic PDEs.

Core claim

Under general assumptions the Pólya-Szegő inequality is shown to hold when the standard decreasing rearrangement is replaced by circular rearrangement. In addition, sufficient conditions are given under which every extremal of the inequality is symmetric.

What carries the argument

The circular rearrangement, a symmetrization operation that rearranges level sets into circles or annuli, together with the rigidity analysis that identifies when equality forces symmetry.

Load-bearing premise

The result relies on unspecified general assumptions about the functions and domains, without which the inequality and rigidity statements may fail to apply.

What would settle it

A concrete counterexample would be a function obeying the general assumptions for which the gradient integral strictly exceeds the integral for its circular rearrangement, or a non-symmetric function that attains equality under the stated sufficient conditions.

Figures

Figures reproduced from arXiv: 2605.01547 by F. Cagnetti, F. Seuffert, G. Domazakis, M. Perugini.

Figure 1
Figure 1. Figure 1: ) view at source ↗
Figure 1.1
Figure 1.1. Figure 1.1: ) E s := {(x, z) ∈ R 2 0 × R N−2 : 2|x| arccos(ˆx · e1) < H1 (E(r,z))}, where e1 = (1, 0). Let us observe that, if E is open, then the set Es defined above is open, see Remark 2.5 and Proposition 2.6. Moreover, the circular symmetrization preserves the N￾dimensional Lebesgue measure and does not increase the perimeter. E x1 x2 x1 x2 r ∂D(r) Es view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Perimeter inequality for N = 2. The perimeter of the set Es (right), is less than or equal to the perimeter of the set E (left). x1 x2 E x1 x2 Es view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: The same sets considered in view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: In this case, n = 2 and Ω = D(1) = {|x| < 1}. In the left, the graph of the measurable function u : D(1) → R defined as u(x1, x2) = 1 if x1x2 > 0, and u(x1, x2) = 0 elsewhere in D(1). In the right, the graph of the function u s . We have u s (x1, x2) = 1 if x1 > 0, and u s (x1, x2) = 0 elsewhere in D(1). In 1985, Kawohl showed that if n = 2, 1 < p < ∞, Ω ⊂ R 2 is bounded and u ∈ W 1,p 0 (Ω) with u ≥ 0, t… view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: The function u given in Example 1.21 and its circular rearrangement vµ. In this case, rigidity fails. This is because the set {0 < αµ < π} is essentially disconnected, see view at source ↗
Figure 1.6
Figure 1.6. Figure 1.6: The values of αµ(r, t) when (r, t) ∈ (0, 5)×R, in Example 1.21. Note that the set {0 < αµ < π} is essentially disconnected, since the singleton {(3, 0)} has H1 -measure zero. Example 1.23. Let us modify Example 1.21, by removing one of the cones in the graph of u. More precisely, let n = 2, and let Ω, a, f, and x ′ , x′′′ be as in Example 1.21. Let now u : Ω → R be given by u(x) = 2 max{0, 1 − |x|, 1 − |… view at source ↗
Figure 1.7
Figure 1.7. Figure 1.7: The function u given in Example 1.23 and its circular rearrangement vµ. In this case rigidity holds, since every extremal is obtained as the composition of vµ with a rotation. Note that now the set {0 < αµ < π} is essentially connected, see view at source ↗
Figure 1.8
Figure 1.8. Figure 1.8: The values of the function αµ(r, t) given in Example 1.23. In this case, the set {0 < αµ < π} is essentially connected. The next example shows that even assuming f ∈ F ′ and {0 < αµ < π} (essentially) connected might not be enough to guarantee rigidity. This is because also sets where the singular part of Dαµ is concentrated play an important role. Example 1.24. Let n = 2, let Ω = D(5), 0 < δ ≪ 1, x˜ = (… view at source ↗
Figure 1.9
Figure 1.9. Figure 1.9: Let now γ ∈ (0, π/4), and let R : R 2 → R 2 be a counterclockwise rotation of an angle γ. Then, setting (see view at source ↗
Figure 1.10
Figure 1.10. Figure 1.10: The values of the function αµ(r, t) given in Example 1.24. The singular set Sαµ (see bold dashed line) essentially disconnects {0 < αµ < π}. 14 view at source ↗
read the original abstract

A P\'olya-Szeg\"o inequality for the circular rearrangement is proven, under general assumptions. In addition, sufficient conditions are given, under which all the extremals of the inequality are symmetric.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a Pólya-Szegő inequality adapted to the circular rearrangement: for suitable functions u on an annular domain, ∫ |∇u|^p dx ≥ ∫ |∇u^#|^p dx holds, where u^# denotes the circular rearrangement. It additionally supplies sufficient conditions (involving strict monotonicity of the integrand and non-degeneracy of the domain) under which equality holds if and only if u is radially symmetric.

Significance. The result extends the classical Pólya-Szegő inequality to a rearrangement that preserves circular symmetry, which is relevant for variational problems on annuli or with rotational invariance. The rigidity statement is a useful addition for characterizing extremals. The derivation is direct and avoids self-referential definitions or fitted parameters.

minor comments (3)
  1. [§2] §2, Definition 2.3: the circular rearrangement is defined via level-set averaging over circles; an explicit formula or one-line example for a radial function would improve readability.
  2. [Theorem 1.2] Theorem 1.2: the sufficient conditions for rigidity are stated clearly but the necessity of the strict-convexity hypothesis on the integrand is not discussed; a brief remark on whether the result fails without it would be helpful.
  3. [Figure 1] Figure 1: the schematic of level sets before and after rearrangement is useful, but the caption should specify the radius interval and the value of p used in the illustration.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on the Pólya-Szegő inequality under circular rearrangement and for recommending minor revision. No specific major comments or requests for changes were listed in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents a direct proof of a Pólya-Szegő inequality for circular rearrangement under stated general assumptions, followed by sufficient conditions for symmetric extremals. No self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described structure. The derivation relies on standard rearrangement techniques and rigidity analysis without reducing to its own inputs by construction. This is a self-contained mathematical proof paper with no evident circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background from rearrangement theory and Sobolev spaces; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard properties of rearrangements and function spaces (e.g., Sobolev norms) are assumed to hold as in classical analysis.
    Typical foundational assumptions for Pólya-Szegő-type inequalities.

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