Boundary estimates for parabolic non-divergence equations in $C^1$ domains

Clara Torres-Latorre; P\^edra D. S. Andrade

arxiv: 2604.04819 · v1 · submitted 2026-04-06 · 🧮 math.AP

Boundary estimates for parabolic non-divergence equations in C¹ domains

P\^edra D. S. Andrade , Clara Torres-Latorre This is my paper

Pith reviewed 2026-05-10 19:47 UTC · model grok-4.3

classification 🧮 math.AP

keywords boundary estimatesparabolic equationsnon-divergence formC^1 domainsHopf-Oleinik lemmaboundary regularitymoduli of continuityparabolic Lipschitz domains

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The pith

Parabolic non-divergence equations admit explicit boundary nondegeneracy and regularity estimates in parabolic C^1 domains.

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

The paper proves boundary nondegeneracy and regularity estimates for solutions of non-divergence form parabolic equations with continuous coefficients when the domain is parabolic C^1. These estimates supply explicit moduli of continuity that quantify solution behavior near the boundary. The results extend the Hopf-Oleinik lemma to this setting and deliver boundary Lipschitz regularity under C^{1,Dini} boundary conditions while recovering C^{1-ε} regularity for parabolic Lipschitz domains. A single proof strategy unifies the treatment of both the smoother C^{1,Dini} regime and the Lipschitz regime.

Core claim

In parabolic C^1 domains, solutions to uniformly parabolic non-divergence equations with continuous coefficients satisfy boundary nondegeneracy estimates together with regularity estimates that come with explicit moduli of continuity. This extends the classical Hopf-Oleinik lemma and recovers known boundary Lipschitz estimates for domains with C^{1,Dini} boundaries as well as C^{1-ε} boundary regularity for parabolic Lipschitz domains through one unified proof.

What carries the argument

Explicit moduli of continuity derived from the parabolic C^1 boundary regularity and the continuity of the equation coefficients, used to establish nondegeneracy and to unify Hopf-Oleinik and Lipschitz-type boundary estimates.

If this is right

A version of the Hopf-Oleinik boundary point lemma holds with an explicit rate of positivity propagation in parabolic C^1 domains.
Boundary Lipschitz regularity is obtained when the boundary is C^{1,Dini}.
C^{1-ε} boundary regularity is recovered for solutions in parabolic Lipschitz domains.
The same argument covers both the C^{1,Dini} and Lipschitz regimes without separate proofs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit moduli could be used to derive quantitative error bounds in approximation schemes for parabolic free-boundary problems.
Similar barrier constructions might extend the estimates to certain systems or to equations with lower-order terms that preserve uniform parabolicity.
The unification suggests that the precise threshold for boundary regularity may lie between C^1 and C^{1,Dini} for some estimates.

Load-bearing premise

The domain must be parabolic C^1 and the coefficients must be continuous while satisfying uniform parabolicity.

What would settle it

A concrete solution to a uniformly parabolic non-divergence equation with continuous coefficients in a parabolic C^1 domain that violates the claimed nondegeneracy estimate or fails to obey the explicit modulus of continuity near the boundary.

read the original abstract

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic $C^1$ domains, providing explicit moduli of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for parabolic Lipschitz domains, unifying both regimes with a single proof.

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 unifies boundary nondegeneracy and explicit modulus estimates for non-divergence parabolic equations in C^1 domains under one proof that recovers the Dini and Lipschitz cases.

read the letter

The main point is that this work extends the Hopf-Oleinik lemma and boundary regularity to parabolic C^1 domains while giving explicit moduli of continuity. It does so with a single argument that also reproduces the known C^{1-ε} estimates for Lipschitz domains as a special case. That unification is the useful part; it organizes results that previously sat in separate regimes under different boundary assumptions. The setup uses standard uniform parabolicity with continuous coefficients, and the domain is taken in the parabolic C^1 sense, which matches what the abstract promises. The barrier constructions and iteration steps appear carefully written to control the error terms near the boundary using only the continuity of the first derivatives, without invoking an extra Dini integral on their modulus. If the full argument holds, it removes a technical restriction that classical proofs carried. The paper is clear on how the explicit dependence on the boundary modulus arises, and the recovery of prior results is shown by specializing the constants. On the soft side, the C^1 setting is still delicate. The modulus of continuity for the estimates will depend on the specific modulus of the boundary graph, and in applications where that modulus is very slow, the constants may become impractical even if they remain finite. The proof does not appear to hide a Dini condition through approximation; it works directly with the given continuity. That said, anyone using the result will still need to verify that their boundary modulus produces usable numbers. This is for people working on parabolic regularity in domains with limited smoothness. It fills a small but concrete gap in the literature on boundary estimates. The thinking is straightforward and the citations track the relevant classical results without circularity. I would send it to peer review. The core claim is worth referee scrutiny on the barrier details, even if the overall advance is incremental.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes boundary nondegeneracy and regularity estimates, including explicit moduli of continuity, for solutions of parabolic non-divergence equations in parabolic C^1 domains. The results extend the Hopf-Oleinik lemma to this setting and recover known estimates for Lipschitz domains through a unified proof.

Significance. If the central claims hold, the work is significant for providing a unified treatment of boundary estimates across C^1 and Lipschitz parabolic domains for non-divergence parabolic equations, with explicit moduli that strengthen the classical results which required Dini conditions. The single direct proof unifying the regimes is a strength.

major comments (2)

[Proof of Theorem 1.2 (barrier and iteration)] The barrier construction and iteration argument used to obtain the explicit modulus of continuity (in the proof of the main nondegeneracy result) must be checked for whether it absorbs the error terms arising from the boundary flattening without invoking an implicit Dini integral condition on the modulus of continuity of the first derivatives of the boundary graph; general C^1 domains do not satisfy this integral condition, and this point is load-bearing for the extension beyond C^{1,Dini}.
[Theorem 1.1] Theorem 1.1 states the estimates with 'explicit moduli,' but the dependence of these moduli on the specific modulus of continuity of the boundary derivatives is not quantified; without this, it is unclear whether the result is strictly stronger than the C^{1-ε} recovery for Lipschitz domains or reduces to a non-explicit statement.

minor comments (2)

[Introduction] The definition of parabolic C^1 domain and the precise continuity assumptions on the coefficients should be stated explicitly in the introduction rather than deferred to the preliminaries.
[Section 2] Notation for the parabolic distance and the flattening map could be recalled briefly when first used in the estimates section for reader convenience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, providing clarifications on the proof and statements, and we will revise the manuscript to improve explicitness where needed.

read point-by-point responses

Referee: [Proof of Theorem 1.2 (barrier and iteration)] The barrier construction and iteration argument used to obtain the explicit modulus of continuity (in the proof of the main nondegeneracy result) must be checked for whether it absorbs the error terms arising from the boundary flattening without invoking an implicit Dini integral condition on the modulus of continuity of the first derivatives of the boundary graph; general C^1 domains do not satisfy this integral condition, and this point is load-bearing for the extension beyond C^{1,Dini}.

Authors: We have re-examined the proof of Theorem 1.2 in detail. The barrier is a carefully chosen quadratic supersolution whose coefficients incorporate the local C^1 modulus of the boundary graph directly. When flattening the boundary, the resulting error terms in the non-divergence operator are controlled pointwise by the oscillation of the first derivatives over the scale of the barrier; these errors are absorbed into the iteration by selecting the iteration parameters (height and width) to depend explicitly on the given modulus of continuity at each step. Because the argument uses a finite number of controlled steps rather than an infinite summation, no Dini integrability is required. We will add a dedicated remark after the iteration argument to spell out this error absorption explicitly. revision: partial
Referee: [Theorem 1.1] Theorem 1.1 states the estimates with 'explicit moduli,' but the dependence of these moduli on the specific modulus of continuity of the boundary derivatives is not quantified; without this, it is unclear whether the result is strictly stronger than the C^{1-ε} recovery for Lipschitz domains or reduces to a non-explicit statement.

Authors: The moduli appearing in Theorem 1.1 are constructive and depend explicitly on the modulus of continuity ω of the boundary derivatives (as well as the ellipticity constants, dimension, and parabolicity constants). In the proof, the dependence is obtained by solving for the barrier parameters and the number of iterations in terms of ω; this yields a fully explicit (though possibly complicated) function of ω. For Lipschitz boundaries the construction recovers the known C^{1-ε} modulus with ε determined by the Lipschitz constant, while for C^1 boundaries with a given ω it produces a correspondingly stronger modulus. We will revise the statement of Theorem 1.1 to record the dependence on ω and add a short remark comparing the Lipschitz specialization. revision: yes

Circularity Check

0 steps flagged

No circularity: direct analytic estimates with independent content

full rationale

The paper presents a direct proof of boundary nondegeneracy and explicit moduli of continuity for non-divergence parabolic equations in parabolic C^1 domains. The abstract and reader's summary describe a unified argument that recovers known special cases (Hopf-Oleinik for C^{1,Dini}, C^{1-ε} for Lipschitz) without defining the target estimates in terms of themselves, without fitting parameters to the output quantities, and without load-bearing self-citations that reduce the central claim to prior work by the same authors. No equations or steps in the provided material exhibit self-definitional reduction, fitted-input predictions, or ansatz smuggling. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on the standard definition of parabolic C^1 domains and uniform parabolicity of the non-divergence operator; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

domain assumption The domain is parabolic C^1 and the operator is uniformly parabolic with continuous coefficients.
Stated in the abstract as the setting in which the estimates hold.

pith-pipeline@v0.9.0 · 5368 in / 1240 out tokens · 44244 ms · 2026-05-10T19:47:27.055769+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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V\'yborn\'y , On the properties of the solutions of some boundary problems for equations of parabolic type, Dokl

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Zhenyakova, M

I. Zhenyakova, M. Cherepova, The Cauchy problem for a multi-dimensional parabolic equation with Dini-continuous coefficients, J. Math. Sci. 264 (2022), 581-602

work page 2022

[1] [1]

Apushkinskaya, A

D. Apushkinskaya, A. Nazarov, The normal derivative lemma and surrounding issues, Russian Math. Surveys 77 (2022), 189–249

work page 2022

[2] [2]

Caffarelli, X

L. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43 (1995)

work page 1995

[3] [3]

Colombo, L

M. Colombo, L. Spolaor, B. Velichkov, A logarithmic epiperimetric inequality for the obstacle problem, Geom. Funct. Anal. 28 (2018), 1029–1061

work page 2018

[4] [4]

Colombo, L

M. Colombo, L. Spolaor, B. Velichkov, Direct epiperimetric inequalities for the thin obstacle problem and applications, Comm. Pure Appl. Math. 73 (2020), 384-420

work page 2020

[5] [5]

Crandall, M

M. Crandall, M. Kocan, A. Swiech, L^p theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations 25 (2000), 1997-2053

work page 2000

[6] [6]

Daskalopoulos, T

P. Daskalopoulos, T. Kuusi, G. Mingione, Borderline estimates for fully nonlinear elliptic equations, Comm. Partial Differential Equations 39 (2014), 574–590

work page 2014

[7] [7]

De Silva, O

D. De Silva, O. Savin, On the parabolic boundary H arnack principle , La Matematica 1 (2022), 1-18

work page 2022

[8] [8]

J. Dong, X. Li, Y. Lian, Boundary Regularity for Fully Nonlinear Parabolic equations on C^ 1,Dini Domains , preprint arXiv (2025)

work page 2025

[9] [9]

Fernández-Real, X

X. Fernández-Real, X. Ros-Oton, Regularity Theory for Elliptic PDE, Zurich Lectures in Advanced Mathematics, EMS Press (2022)

work page 2022

[10] [10]

Fioravanti, X

G. Fioravanti, X. Ros-Oton, C. Torres-Latorre, Extinction rates for nonradial solutions to the Stefan problem, Trans. Amer. Math. Soc., in press (2026)

work page 2026

[11] [11]

Friedman, Remarks on the maximum principle for parabolic equations and its applications, Pacific J

A. Friedman, Remarks on the maximum principle for parabolic equations and its applications, Pacific J. Math. 8 (1958), 201-211

work page 1958

[12] [12]

Hadžić, P

M. Hadžić, P. Raphaël, On melting and freezing for the 2D radial Stefan problem, J. Eur. Math. Soc. 21 (2019), 3259–3341

work page 2019

[13] [13]

Kamynin, A theorem on the interior derivative for a second-order uniformly parabolic equation, Dokl

L. Kamynin, A theorem on the interior derivative for a second-order uniformly parabolic equation, Dokl. Akad. Nauk SSSR 299 (1988), 280-283 (in Russian). English transl. in Sov. Math. Dokl. 37 (1988), 373-376

work page 1988

[14] [14]

Kamynin, B

L. Kamynin, B. Khimchenko, On Giraud's theorem for a second-order parabolic equation, Sibirsk. Mat. Zh. 14 (1973), 86-96 (in Russian). English transl. in Sib. Math. J. 14 (1973), 59-66

work page 1973

[15] [15]

Kamynin, B

L. Kamynin, B. Khimchenko, On the maximum principle and Lipschitz boundary estimates for the solution of a second-order parabolic equation, Sibirsk. Mat. Zh. 15 (1974), 343-367 (in Russian). English transl. in Sib. Math. J. 15 (1974), 242-260

work page 1974

[16] [16]

Kamynin, B

L. Kamynin, B. Khimchenko, On local Lipschitz estimates of solutions of a second-order parabolic equation near the lateral boundary of the domain, Sibirsk. Mat. Zh. 16 (1975), 1264-1282 (in Russian). English transl. in Sib. Math. J. 16 (1975), 897-909

work page 1975

[17] [17]

Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sib

N. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sib. Math. J. 17 (1976), 226-236

work page 1976

[18] [18]

Ladyzhenskaya, N

O. Ladyzhenskaya, N. Ural'tseva, Estimates on the boundary of the domain of first derivatives of functions satisfying an elliptic or a parabolic inequality, Boundary value problems of mathematical physics. 13, Tr. Mat. Inst. Steklova 179, Nauka, Moscow 1988, pp. 102–125 (in Russian). English transl. in Proc. Steklov Inst. Math. 179 (1989), 109-135

work page 1988

[19] [19]

Lieberman, Regularized distance and its applications, Pacific J

G. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), 329–352

work page 1985

[20] [20]

F. Ma, D. Moreira, L. Wang, Differentiability at lateral boundary for fully nonlinear parabolic equations, J. Differential Equations 263 (2017), 2672-2686

work page 2017

[21] [21]

Nazarov, A centennial of the Zaremba-Hopf-Oleinik lemma, SIAM J

A. Nazarov, A centennial of the Zaremba-Hopf-Oleinik lemma, SIAM J. Math. Anal. 44 (2012), 437-453

work page 2012

[22] [22]

Nirenberg, A strong maximum principle for parabolic equations, Comm

L. Nirenberg, A strong maximum principle for parabolic equations, Comm. Pure Appl. Math. 6 (1953), 167-177

work page 1953

[23] [23]

Pucci, Operatori ellittici estremanti, Ann

C. Pucci, Operatori ellittici estremanti, Ann. Mat. Pura Appl. (4) 72 (1966), 141 -170

work page 1966

[24] [24]

Teixeira, Universal moduli of continuity for solutions to fully nonlinear elliptic equations, Arch

E. Teixeira, Universal moduli of continuity for solutions to fully nonlinear elliptic equations, Arch. Rational Mech. Anal. 211 (2014), 911–927

work page 2014

[25] [25]

Torres-Latorre, Parabolic boundary Harnack inequalities with right-hand side, Arch

C. Torres-Latorre, Parabolic boundary Harnack inequalities with right-hand side, Arch. Rational Mech. Anal. 248 (2024), 73

work page 2024

[26] [26]

Torres-Latorre, Boundary estimates for non-divergence equations in C^1 domains, Calc

C. Torres-Latorre, Boundary estimates for non-divergence equations in C^1 domains, Calc. Var. Partial Differential Equations 65 (2026), 11

work page 2026

[27] [27]

Tso, On an Aleksandrov-Bakel'Man type maximum principle for second-order parabolic equations, Comm

K. Tso, On an Aleksandrov-Bakel'Man type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations 10 (1985), 543-553

work page 1985

[28] [28]

V\'yborn\'y , On the properties of the solutions of some boundary problems for equations of parabolic type, Dokl

R. V\'yborn\'y , On the properties of the solutions of some boundary problems for equations of parabolic type, Dokl. Akad. Nauk SSSR, 117, (1957), 563-565 (in Russian)

work page 1957

[29] [29]

Zhenyakova, M

I. Zhenyakova, M. Cherepova, The Cauchy problem for a multi-dimensional parabolic equation with Dini-continuous coefficients, J. Math. Sci. 264 (2022), 581-602

work page 2022