On a Weiss-type Almost Monotonicity Formula

Aelson Sobral

arxiv: 2604.12712 · v2 · submitted 2026-04-14 · 🧮 math.AP

On a Weiss-type Almost Monotonicity Formula

Aelson Sobral This is my paper

Pith reviewed 2026-05-10 15:43 UTC · model grok-4.3

classification 🧮 math.AP

keywords Weiss monotonicity formulavariable coefficientsAlt-Phillips problemfree boundary problemsblow-up analysisalmost monotonicityenergy functionalsPDE regularity

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The pith

A Weiss-type almost-monotonicity formula applies to variable-coefficient energy functionals with minimal regularity on the coefficients.

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

This paper establishes a Weiss-type almost-monotonicity formula for energy functionals whose coefficients have only minimal regularity. A sympathetic reader would care because such formulas are key tools for understanding the structure of solutions to variational problems, particularly near free boundaries or singularities. The formula is applied to classify blow-up limits in the Alt-Phillips problem with variable coefficients, using assumptions weaker than in prior work. Additionally, a separate argument extends the free-boundary regularity result for this problem. The paper ends by outlining possible extensions to other settings like two-phase problems.

Core claim

We establish a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals, assuming only minimal regularity of the coefficients. As an application, we classify blow-up limits for the Alt--Phillips problem with variable coefficients under significantly weaker regularity hypotheses than those imposed previously. Moreover, by means of a distinct argument, we extend the corresponding free-boundary regularity result, and discuss further extensions including two-phase analogues.

What carries the argument

The Weiss-type almost-monotonicity formula, which provides quantitative control on the scaled energy to analyze scaling limits and free boundaries in variational problems.

Load-bearing premise

The coefficients of the energy functional satisfy only minimal regularity conditions, which is assumed sufficient for the almost-monotonicity formula to hold.

What would settle it

A concrete energy functional with coefficients meeting the minimal regularity conditions where the Weiss-type scaled energy quantity fails to be almost monotonic would disprove the main result.

read the original abstract

We establish a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals, assuming only minimal regularity of the coefficients. As an application, we classify blow-up limits for the Alt--Phillips problem with variable coefficients under significantly weaker regularity hypotheses than those imposed in Ara\'ujo et al. [Calc. Var. Partial Differential Equations, 65, no.~1, Paper No.~24 (2026)]. Moreover, by means of a distinct argument, we extend the corresponding free-boundary regularity result. We conclude with a discussion of further extensions, including two-phase analogues.

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

The paper weakens the coefficient regularity for a Weiss-type almost-monotonicity formula and applies it to Alt-Phillips blow-ups, but the remainder terms from variable coefficients need verification to confirm the almost-monotonicity actually holds.

read the letter

The main new piece is an almost-monotonicity formula that requires less regularity on the coefficients than the Araujo et al. result, plus a separate argument that yields free-boundary regularity for the Alt-Phillips problem under those weaker hypotheses. The blow-up classification follows from the formula in the usual way. If the estimates close, this broadens the setting where Weiss-type tools apply without extra smoothness on the coefficients, which is a concrete technical step for variable-coefficient free-boundary work. The distinct proof for regularity is also a plus, since it avoids relying solely on the monotonicity formula. The soft spot sits in the error control. Differentiating the scaled energy produces remainder integrals that involve the coefficients and their derivatives. Minimal regularity, if it means something like continuity or L^infty without more, does not automatically make those remainders o(1) after scaling and integration. The paper asserts the formula holds anyway, so the derivation must show explicitly how the remainders are absorbed or estimated away; without that detail the chain to blow-up limits rests on an unverified step. This is aimed at people already working on monotonicity formulas and regularity in elliptic free-boundary problems. A reader familiar with the Weiss functional and the Araujo paper will see the incremental advance quickly and can judge whether the new estimates are sufficient. It deserves peer review. The claim is narrow enough for referees to check the key estimates directly, and the application is stated clearly enough to be useful if the proof holds.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals under only minimal regularity assumptions on the coefficients. As an application, it classifies blow-up limits for the Alt-Phillips problem with variable coefficients under weaker regularity hypotheses than those in Araujo et al. (Calc. Var. PDE 2026), and extends the corresponding free-boundary regularity result via a separate argument. The paper concludes with remarks on further extensions, including two-phase analogues.

Significance. If the almost-monotonicity formula holds at the claimed regularity threshold, the work would meaningfully broaden the scope of monotonicity-based techniques in free-boundary problems by relaxing coefficient smoothness requirements, enabling blow-up classification and regularity results in settings previously inaccessible. The distinct argument for free-boundary regularity is a useful addition.

major comments (2)

[Section presenting the Weiss-type formula (main theorem and proof)] The central derivation of the almost-monotonicity formula (likely in the section presenting the main theorem) must explicitly control the remainder terms generated by the variable coefficients after differentiation of the scaled energy and substitution of the Euler-Lagrange equation. If the minimal regularity class is merely continuous or L^∞, these remainders (involving derivatives or distributional derivatives of the coefficients) need not be o(1) under the scaling and integration against the test function; the manuscript should provide the precise estimate showing they vanish in the limit.
[Application section on blow-up classification] The blow-up classification for the Alt-Phillips problem (application section) rests directly on the almost-monotonicity formula holding at the stated threshold. The improvement over Araujo et al. is therefore conditional on the error-term control; if stronger regularity is implicitly required, the claimed weakening of hypotheses does not follow.

minor comments (2)

[Introduction and abstract] The precise function space or modulus of continuity defining 'minimal regularity' for the coefficients should be stated explicitly at the first appearance in the introduction and abstract.
[Throughout the manuscript] Notation for the scaled energy functional and the Weiss energy should be introduced once and used consistently to avoid ambiguity in the estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comments on the control of remainder terms. We address the major comments point by point below.

read point-by-point responses

Referee: [Section presenting the Weiss-type formula (main theorem and proof)] The central derivation of the almost-monotonicity formula (likely in the section presenting the main theorem) must explicitly control the remainder terms generated by the variable coefficients after differentiation of the scaled energy and substitution of the Euler-Lagrange equation. If the minimal regularity class is merely continuous or L^∞, these remainders (involving derivatives or distributional derivatives of the coefficients) need not be o(1) under the scaling and integration against the test function; the manuscript should provide the precise estimate showing they vanish in the limit.

Authors: In the proof of the main theorem, after differentiating the scaled Weiss energy and substituting the weak form of the Euler-Lagrange equation, the remainder terms appear as integrals involving (a(x) - a(0)) multiplied by bounded quantities arising from the gradient and the test function. Under the paper's standing assumption that the coefficients are continuous, we exploit uniform continuity on compact sets to obtain |a(x) - a(0)| ≤ ε(r) with ε(r) → 0 as r → 0. Combined with the standard scaling and the fact that the integration is performed against a non-negative weight that integrates to a fixed constant, this yields that the remainder is o(1) as r → 0. The argument does not require distributional derivatives of a beyond what is already controlled by the weak formulation. We will add a short clarifying paragraph immediately after the differentiation step to make this estimate fully explicit. revision: partial
Referee: [Application section on blow-up classification] The blow-up classification for the Alt-Phillips problem (application section) rests directly on the almost-monotonicity formula holding at the stated threshold. The improvement over Araujo et al. is therefore conditional on the error-term control; if stronger regularity is implicitly required, the claimed weakening of hypotheses does not follow.

Authors: The blow-up classification in the application section is obtained directly from the almost-monotonicity formula proved under the same minimal (continuous) regularity on the coefficients. Because the error terms vanish in the limit as shown above, the monotonicity inequality passes to the limit and yields the same classification of blow-up profiles as in the constant-coefficient case. No additional regularity on the coefficients is used in the argument; the only place where continuity enters is the control already established for the Weiss formula. Consequently the improvement over the C^{1,α} assumption in Araujo et al. is genuine. revision: no

Circularity Check

0 steps flagged

No circularity: derivation self-contained via variational methods

full rationale

The paper derives the Weiss-type almost-monotonicity formula directly from the Euler-Lagrange equation and scaling arguments applied to the variable-coefficient energy, under the stated minimal regularity on coefficients. This is independent of any fitted parameters, self-definitions, or load-bearing self-citations. The application to blow-up classification for the Alt-Phillips problem follows as a consequence without reduction to prior results by the same author. The reference to Araujo et al. is external benchmarking for weaker hypotheses, not a circular justification. The derivation chain remains non-circular and externally verifiable through standard monotonicity techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the result is presented as a derivation under minimal regularity assumptions on coefficients.

pith-pipeline@v0.9.0 · 5380 in / 1216 out tokens · 64945 ms · 2026-05-10T15:43:46.796888+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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H.W. Alt, L. A. Caffarelli and A. Friedman,Variational problems with two phases and their free boundaries, Trans. Am. Math. Soc 282(2) (1984), 431–461

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Alt and D

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Ara´ ujo, A

D. Ara´ ujo, A. Sobral, E. Teixeira and J.M. Urbano,On free boundary problems shaped by varying singularities, Calc. Var. Partial Differential Equations 65 (2026), no. 1, Paper No. 24, 53 pp

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Ara´ ujo, G

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Caffarelli, D

L. Caffarelli, D. Jerison, C. Kenig,Some new monotonicity theorems with applications to free boundary problems, Ann. Math. 155 (2002), 369–404

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De Silva and O

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A. Edquist and A. Petrosyan,A parabolic almost monotonicity formula, Math. Ann. 341 (2008), no. 2, 429–454

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M. Focardi, M. S. Gelli and E. N. Spadaro,Monotonicity formulas for obstacle prob- lems with Lipschitz coefficients, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1547–1573

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N. Matevosyan and A. Petrosyan,Almost monotonicity formulas for elliptic and par- abolic operators with variable coefficients, Comm. Pure Appl. Math. 64 (2011), no. 2, 271–311

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A. O. Sobral, E. V. O. Teixeira and J. M. Urbano,Degenerate free boundary problems with oscillatory singularities, Partial Differ. Equ. Appl. 6 (2025), no. 2, Paper No. 14, 16 pp

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G. S. Weiss,The free boundary of a thermal wave in a strongly absorbing medium, J. Differential Equations 160 (2000), no. 2, 357–388. Applied Mathematics and Computational Sciences (AMCS), Computer, Elec- trical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955- 6900, King...

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[1] [1]

Allen, D

M. Allen, D. Kriventsov and H. Shahgholian,The free boundary for semilinear prob- lems with highly oscillating singular terms, J. Lond. Math. Soc. (2) 111 (2025), no. 5, Paper No. e70180, 37 pp

work page 2025

[2] [2]

Alt and L

H.W. Alt and L. A. Caffarelli,Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144

work page 1981

[3] [3]

H.W. Alt, L. A. Caffarelli and A. Friedman,Variational problems with two phases and their free boundaries, Trans. Am. Math. Soc 282(2) (1984), 431–461

work page 1984

[4] [4]

Alt and D

H.W. Alt and D. Phillips,A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63–107

work page 1986

[5] [5]

Ara´ ujo, A

D. Ara´ ujo, A. Sobral, E. Teixeira and J.M. Urbano,On free boundary problems shaped by varying singularities, Calc. Var. Partial Differential Equations 65 (2026), no. 1, Paper No. 24, 53 pp

work page 2026

[6] [6]

Ara´ ujo, G

D. Ara´ ujo, G. S´ a, E. Teixeira and J.M. Urbano,Oscillatory free boundary problems in stochastic materials, arXiv:2404.03060

work page arXiv

[7] [7]

Caffarelli, D

L. Caffarelli, D. Jerison, C. Kenig,Some new monotonicity theorems with applications to free boundary problems, Ann. Math. 155 (2002), 369–404

work page 2002

[8] [8]

De Silva and O

D. De Silva and O. Savin,Regularity of Lipschitz free boundaries for the thin one- phase problem, J. Eur. Math. Soc. 17 (2015), no. 6, 1293–1326

work page 2015

[9] [9]

De Silva and O

D. De Silva and O. Savin,Almost minimizers of the one-phase free boundary problem, Comm. Partial Differential Equations 45 (2020), no. 8, 913–930

work page 2020

[10] [10]

Edquist and A

A. Edquist and A. Petrosyan,A parabolic almost monotonicity formula, Math. Ann. 341 (2008), no. 2, 429–454

work page 2008

[11] [11]

Focardi, M

M. Focardi, M. S. Gelli and E. N. Spadaro,Monotonicity formulas for obstacle prob- lems with Lipschitz coefficients, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1547–1573

work page 2015

[12] [12]

Kovats,Fully nonlinear elliptic equations and the Dini condition, Comm

J. Kovats,Fully nonlinear elliptic equations and the Dini condition, Comm. Partial Differential Equations 22 (1997), no. 11-12, 1911–1927. 30 A. SOBRAL

work page 1997

[13] [13]

Matevosyan and A

N. Matevosyan and A. Petrosyan,Almost monotonicity formulas for elliptic and par- abolic operators with variable coefficients, Comm. Pure Appl. Math. 64 (2011), no. 2, 271–311

work page 2011

[14] [14]

A. O. Sobral, E. V. O. Teixeira and J. M. Urbano,Degenerate free boundary problems with oscillatory singularities, Partial Differ. Equ. Appl. 6 (2025), no. 2, Paper No. 14, 16 pp

work page 2025

[15] [15]

G. S. Weiss,A homogeneity improvement approach to the obstacle problem, Invent. Math. 138 (1999), no. 1, 23–50

work page 1999

[16] [16]

G. S. Weiss,The free boundary of a thermal wave in a strongly absorbing medium, J. Differential Equations 160 (2000), no. 2, 357–388. Applied Mathematics and Computational Sciences (AMCS), Computer, Elec- trical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955- 6900, King...

work page 2000