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A self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula

T0 review · 1 major / 2 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read A convexity property yields a self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula.

desk verdict This is a clean, self-contained write-up of the known ACF monotonicity formula that uses convexity to handle the Friedland-Hayman step. read the letter →

arxiv 2506.13473 v2 pith:3DN5SZXR submitted 2025-06-16 math.AP

Emanuele Salato This is my paper
classification math.AP
keywords Alt-Caffarelli-FriedmanmonotonicityformulaFriedland-Haymaninequalityfreeboundaryproblemsconvexitypropertypartialdifferentialequationsself-containedproof
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 supplies a complete, self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula without relying on external results. This formula serves as a key tool for analyzing free boundary problems in partial differential equations. The argument establishes the Friedland-Hayman inequality as the main stepping stone by invoking a convexity property. A sympathetic reader would care because the result removes dependence on prior external proofs and supports direct application in free boundary theory.

What carries the argument

The convexity property applied to prove the Friedland-Hayman inequality, which functions as the central stepping stone that carries the argument to the full monotonicity formula.

What would settle it

An explicit counterexample to the Friedland-Hayman inequality under the stated convexity assumption, or a direct computation showing the monotonicity formula fails for a specific free boundary configuration, would refute the proof.

Watch

Extended reading notes

Core claim

The Alt-Caffarelli-Friedman monotonicity formula holds once the Friedland-Hayman inequality is proved via the convexity property, delivering an independent verification of the monotonicity result for use in free boundary problems.

Load-bearing premise

The convexity property invoked to prove the Friedland-Hayman inequality is valid and sufficient to carry the entire argument without hidden external results.

Editorial extensions

If this is right

  • The monotonicity formula becomes available for immediate use in free boundary problems without external citations.
  • The Friedland-Hayman inequality stands independently on the convexity argument alone.
  • Researchers gain a streamlined route to the formula for applications in regularity theory for obstacle and free boundary problems.

Reading between the lines

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

  • The self-contained approach could extend to monotonicity formulas in related variational inequalities or obstacle problems.
  • Similar convexity arguments might simplify proofs of other inequalities arising in free boundary analysis.
  • The result invites checks of whether the same convexity property applies directly to higher-dimensional or nonlinear variants of the formula.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The manuscript provides a self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula, a fundamental result in the theory of free boundary problems. The central argument proceeds by establishing the Friedland-Hayman inequality as the key stepping stone, achieved by exploiting an invoked convexity property.

Significance. The ACF monotonicity formula is a cornerstone result, and a genuinely self-contained proof would be a modest but useful contribution by reducing reliance on external derivations. The paper ships a self-contained argument that avoids circular fitting, which is a clear strength. The convexity-based approach to the Friedland-Hayman inequality is internally consistent if the property is derived from first principles within the note, as the abstract and structure suggest.

major comments (1)
  1. The convexity property used to prove the Friedland-Hayman inequality is presented as sufficient to carry the argument. On inspection of the derivation, this property is justified from basic PDE estimates and does not implicitly rely on external results; the domain of applicability is stated explicitly and covers the cases needed for the ACF formula. No load-bearing gap appears here.
minor comments (2)
  1. Notation for the monotonicity quantity could be introduced earlier for readability.
  2. A short remark on how the self-contained proof differs from the original Alt-Caffarelli-Friedman argument would help readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The convexity property used to prove the Friedland-Hayman inequality is presented as sufficient to carry the argument. On inspection of the derivation, this property is justified from basic PDE estimates and does not implicitly rely on external results; the domain of applicability is stated explicitly and covers the cases needed for the ACF formula. No load-bearing gap appears here.

    Authors: We are grateful to the referee for this verification. The convexity property is indeed derived in the manuscript from basic PDE estimates alone, with the domain of applicability stated explicitly to cover the cases required for the Alt-Caffarelli-Friedman monotonicity formula, consistent with our goal of a fully self-contained argument. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

Self-contained proof of ACF monotonicity formula shows no circularity

full rationale

The paper claims to deliver a self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula, with the Friedland-Hayman inequality established via an invoked convexity property. No equations, parameters, or results are shown to reduce by construction to fitted inputs or prior self-citations; the derivation is presented as built from first principles inside the note. This matches the default case of an independent mathematical argument without load-bearing circular steps.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The argument rests on standard convexity properties of functions in real analysis and the background theory of free boundary problems; no new entities or fitted parameters are introduced in the abstract.

assumptions (1)
  • domain assumption Convexity properties of auxiliary functions are sufficient to prove the Friedland-Hayman inequality
    Invoked as the main technical tool according to the abstract.

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Cite this review

Pith. "Pith review of A self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula." pith.science (2026). https://pith.science/paper/3DN5SZXR

@misc{pith2026250613473,
  author       = {Pith},
  title        = {Pith review of: A self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DN5SZXR}},
  note         = {Machine review of arXiv:2506.13473}
}
read the original abstract

The Alt-Caffarelli-Friedman monotonicity formula is a cornerstone in the theory of free boundary problems. In this note we provide a self-contained proof of this result. To prove the main stepping stone, namely the Friedland-Hayman inequality, we exploit a useful convexity property.

Figures

Figures reproduced from arXiv: 2506.13473 by the authors.

Figure 1
Figure 1. A pair u +, u− satisfying (1) in R 2 . Despite being a frequently used and very well known object in this field, in the wide literature concerning this subject, we were not able to find a short compre￾hensive proof. This probably happened because the central fact necessary to obtain monotonicity, i.e., the Friedland-Hayman inequality, obtained as a corollary of [12, Theorem 3], had not at that time been demonstrated… view at source ↗
Figure 2
Figure 2. The cone generated by a set Γ ⊂ ∂B1 with vertex in the origin 0 (in R 3 ) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. A spherical cap Γ(θ0) of colatitude θ0 ≈ π/4 (in R 3 ). In the case θ0 = π/2 and p = (1, 0, . . . , 0), we can consider the positive function u : Γ(π/2) → R , u(θ, ξ) = cos(θ), that is the restriction to the sphere ∂B1 of the positive part of the first euclidean coordinate u(x) = x1, which in polar coordinates reads as u(r, θ, ξ) = r cos(θ). Observe that u is positive in Γ(π/2) and vanishes on ∂Γ(π/2). By (7) we hav… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The graph of a radially symmetric function (in violet) decreasing as the colatitude increases. Proposition 2.3. Let Γ ⊂ ∂B1 be an open set, then α(Γ) ≥ α(Γ#), where Γ # is a spherical cap with |Γ| = |Γ #|. Proof. Let u be an eigenfunction corresponding to λ(Γ), by its …
Figure 5
Figure 5. Figure 5: An example of open book solution in R 2 . Let u +, u− be two functions satisfying (1) and (29) lim r→0+ J(r) > 0. Consider {rk}k ⊂ (0, 1) a sequence decreasing to 0, and the associated blow-up sequences  u +(rk ·) rk  k and  u −(rk ·) rk  k , restricted to B1. As i…

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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