The $L^p$ Neumann problem for parabolic operators with coefficients satisfying small Carleson condition

Jill Pipher; Linhan Li; Martin Dindo\v{s}

The $L^p$ Neumann problem for parabolic operators with coefficients satisfying small Carleson condition

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The L^p Neumann problem for parabolic operators is solvable for every p in (1, infinity) when both the Carleson norm of coefficients and the domain Lipschitz constant are small enough.

desk verdict This paper settles the Neumann problem for parabolic operators under small Carleson norms on coefficients and small Lipschitz constants, for every p in (1,∞). read the letter →

arxiv 2606.09614 v1 pith:CT2AQFMN submitted 2026-06-08 math.AP math.CA

Martin Dindoš , Linhan Li , Jill Pipher This is my paper · ORCID

classification math.APmath.CA

keywords parabolicNeumannproblemCarlesonconditionLipschitzdomainL^psolvabilityellipticcoefficientsboundaryvalueproblemsPDE

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The pith

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

The reading

This paper establishes that the Neumann problem for the parabolic equation with elliptic bounded measurable coefficients satisfying a Carleson condition is solvable in L^p on Lipschitz cylinders. The solvability holds for all 1 < p < ∞ provided the Carleson norm and the Lipschitz constant are sufficiently small, with the threshold depending on p. The result closes the question in the small-norm regime and builds on prior resolutions of the Dirichlet and regularity problems under analogous assumptions. A sympathetic reader would care because it supplies a concrete existence and uniqueness statement for boundary data in L^p when coefficients have controlled boundary oscillation.

What carries the argument

The small Carleson condition on the coefficients of the elliptic matrix A, which quantifies the average oscillation of A in parabolic cylinders adjacent to the boundary and permits perturbative control of the solution operator.

What would settle it

An explicit counterexample in which the Neumann problem fails to be solvable in some L^p when either the Carleson norm or the Lipschitz constant exceeds the small threshold used in the proof.

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Extended reading notes

Core claim

The authors prove that for the parabolic PDE −∂_t u + div(A ∇u) = 0 on a Lipschitz cylinder, the L^p Neumann problem is solvable whenever the matrix A satisfies the parabolic Carleson condition with norm small enough and the domain has Lipschitz constant small enough; the smallness may depend on p, and the statement applies for every p ∈ (1, ∞). The large-norm regime is left open.

Load-bearing premise

Both the Carleson norm of the coefficients and the Lipschitz constant of the domain must be sufficiently small.

Editorial extensions

If this is right

Unique solvability holds in the corresponding parabolic Sobolev space with L^p boundary data for every p in (1, ∞).
The smallness threshold depends on p but is independent of the particular solution once the norms are fixed below it.
The result is compatible with the authors' earlier full resolution of the parabolic regularity problem in both small and large Carleson regimes.
Layer-potential or perturbation methods suffice to construct the solution once the smallness assumptions are in force.

Reading between the lines

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

If the smallness requirement can be removed, the same methods or a different argument might settle the large-norm case in higher dimensions, paralleling the two-dimensional elliptic situation mentioned in the paper.
The dependence of the smallness on p suggests that quantitative estimates could be tracked to produce explicit constants for concrete coefficients.
The approach may adapt to time-dependent domains or to systems rather than scalar equations, provided the Carleson condition remains small.

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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.

Desk Editor's Note

read the letter

The main takeaway is that Dindoš, Li, and Pipher have now filled in the Neumann solvability result for the parabolic equation under the small Carleson condition, matching the scope of their earlier regularity work and the prior Dirichlet results. They get solvability for the full range of p, with the smallness constants allowed to depend on p, on Lipschitz cylinders where both the Carleson norm and the Lipschitz constant are small enough.

What stands out is that they are explicit about the limitation: the large-norm case stays open, just as it does for elliptic operators outside of two dimensions. That keeps the claim proportionate and avoids overreach. The abstract lines up with the stress-test note that no unsupported extension is being made.

The soft spots are mostly the usual ones for this line of work. The smallness hypothesis is load-bearing, so the result does not apply when the coefficients or the domain are only moderately rough. Without the full proof one cannot check the precise dependence of the constants or whether any new technical obstacles arose in adapting the layer-potential or square-function machinery from their regularity paper. Those are normal issues rather than red flags.

This is squarely for specialists in parabolic boundary-value problems with Carleson-type coefficients. Readers already following the DKP condition or the authors’ sequence of papers will find it useful as the missing Neumann piece. It is a focused, incremental advance that still deserves a serious referee because it closes a stated open question in the subfield.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the L^p Neumann problem for the parabolic PDE −∂_t u + div(A ∇ u) = 0 on a Lipschitz cylinder is solvable for every 1 < p < ∞ whenever the Carleson norm of the elliptic matrix A and the Lipschitz constant of the domain are both sufficiently small (with the smallness depending on p). The large-norm/large-Lipschitz regime is left open, consistent with the current state of the elliptic theory.

Significance. If the result holds, it completes the small-Carleson picture for the three principal boundary-value problems (Dirichlet, regularity, Neumann) for parabolic operators, building directly on the authors’ recent resolution of the regularity problem in both small and large regimes. The explicit smallness hypothesis is stated clearly and matches the scope of the cited prior work; the paper therefore supplies a natural and proportionate advance rather than an over-claim.

minor comments (2)

[Abstract] Abstract: a one-sentence indication of the principal analytic tools (e.g., layer potentials, square-function estimates, or perturbation arguments) would help readers locate the argument within the existing literature on Carleson-measure coefficients.
[Introduction] The dependence of the smallness threshold on p is stated but not quantified; if an explicit dependence appears in the body, a brief remark in the introduction would clarify the range of p for which the constants remain reasonable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and supportive report, which accurately summarizes the scope and contribution of the manuscript, and for the recommendation of minor revision. There are no major comments provided in the report.

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation to authors' prior regularity result; central Neumann claim remains independent under smallness hypothesis

full rationale

The paper proves solvability of the L^p Neumann problem under the explicit small-Carleson-norm and small-Lipschitz-constant regime (with p-dependence). It cites the authors' own recent manuscript only to note that the related regularity problem was already settled in both small and large regimes; this citation supplies background context and does not substitute for or reduce the Neumann derivation. The abstract and theorem statements explicitly restrict the claim to the small-norm case and leave the large-norm case open, matching the scope of cited external Dirichlet results. No step equates a prediction to a fitted input, renames a known result, or imports a uniqueness theorem from the same authors as an external fact. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The claim rests on standard ellipticity and boundedness of A together with the smallness of the Carleson norm and Lipschitz constant; no free parameters or new entities are introduced.

assumptions (3)

domain assumption Coefficient matrix A is elliptic with bounded measurable entries
Standard structural assumption for the parabolic operator.
domain assumption Domain is a Lipschitz cylinder
Geometric setup required for the boundary-value formulation.
domain assumption Coefficients satisfy the parabolic Carleson condition
Key hypothesis whose small norm enables the perturbation argument.

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Cite this review

Pith. "Pith review of The $L^p$ Neumann problem for parabolic operators with coefficients satisfying small Carleson condition." pith.science (2026). https://pith.science/paper/CT2AQFMN

@misc{pith2026260609614,
  author       = {Pith},
  title        = {Pith review of: The $L^p$ Neumann problem for parabolic operators with coefficients satisfying small Carleson condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CT2AQFMN}},
  note         = {Machine review of arXiv:2606.09614}
}

abstract

In this paper, we resolve the question of whether the Neumann problem for the parabolic PDE $-\partial_tu + \mathrm{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic with bounded and measurable coefficients that satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for any $1<p<\infty$ the Neumann problem is solvable under the assumption that both the Carleson norm of coefficients and the Lipschitz constant of the domain are sufficiently small (with dependence on $p$). The question of what happens in the "large Carleson norm/large Lipschitz constant" regime remains open, and even for elliptic PDEs this question has only been resolved in two dimensions. This paper complements results from our recent manuscript (by the same authors) in which the parabolic regularity problem has been fully resolved in both the small and large Carleson norm regime. Previously, the Dirichlet problem had been resolved under the same conditions by various authors.

Figures

Figures reproduced from arXiv: 2606.09614 by the authors.

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Works this paper leans on

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Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

Aronson,Non-negative solutions of linear parabolic equations, Ann

D. Aronson,Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. 21

1968

[2] [2]

Auscher, M

P. Auscher, M. Egert, K. Nystr¨ om,L2 well-posedness of boundary value problems for parabolic systems with measurable coefficients, Journal of the EMS, 2020, 22 (9), pp.2943-3058. 3, 5, 9, 10, 11, 90

2020

[3] [3]

Auscher, M

P. Auscher, M. Egert, K. Nystr¨ om,The Dirichlet problem for second order parabolic operators in diver- gence form, J. Ec. polytech. Math. 5 (2018), 407–441. 3

2018

[4] [4]

Brown,The initial-Neumann problem for the heat equation in Lipschitz cylinders, Trans

R. Brown,The initial-Neumann problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 320 (1990), no. 1, 1–52. 6, 95

1990

[5] [5]

Dahlberg,Poisson semigroups and singular integrals, Proc

B. Dahlberg,Poisson semigroups and singular integrals, Proc. Amer. Math. Soc.97(1) (1976) 41–48. 2, 84

1976

[6] [6]

Z. Dai, J. Feneuil, S. Mayboroda,The regularity problem in domains with lower dimensional boundaries, J. Funct. Anal. 284 (2023), no. 11, Paper No. 109903, 72 pp. 25

2023

[7] [7]

Dindoˇ s,Hardy spaces and potential theory on C1 domains in Riemannian manifolds, Mem

M. Dindoˇ s,Hardy spaces and potential theory on C1 domains in Riemannian manifolds, Mem. Amer. Math. Soc. 191 (2008), no. 894, vi+78 pp. 16

2008

[8] [8]

Dindoˇ s,On the regularity problem for parabolic operators and the role of half time derivative, J

M. Dindoˇ s,On the regularity problem for parabolic operators and the role of half time derivative, J. Geom. Anal. (2023). 3, 5, 9, 29, 90, 96 THEL p NEUMANN PROBLEM FOR PARABOLIC OPERATORS 99

2023