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

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.

• The paper establishes Lp Neumann solvability for every 1 < p < ∞ when the coefficient Carleson norm and domain Lipschitz constant are sufficiently small, yielding nontangential gradient estimates controlled by the boundary data.
• The proof combines coefficient smoothing and perturbation, enlarged-cone good-λ estimates, p-adapted square functions, conormal derivative analysis, and sawtooth-domain arguments without requiring block-form or time-independent coefficients.
• The result completes the small-Carleson Dirichlet, Regularity, and Neumann theory for time-varying non-symmetric parabolic operators, while leaving large-Carleson solvability and some half-time-derivative estimates open.

Overview and main result

This paper by Dindoš, Li, and Pipher establishes solvability of the $L^p$ Neumann problem for parabolic operators in divergence form,

$$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$

posed on Lipschitz cylinders $\Omega = \mathcal O \times \mathbb R$, where $\mathcal O$ is a bounded or unbounded Lipschitz domain. The coefficient matrix $A = [a_{ij}(X,t)]$ is assumed uniformly elliptic, bounded, measurable, not necessarily symmetric, and allowed to vary in both space and time. The regularity assumption on $A$ is a Carleson measure condition — the parabolic analog of the DKP condition — requiring that either

$$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$

or the oscillation-based variant $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$ be a Carleson measure with finite norm $\|\mu\|_C$. Here $\delta(X,t)$ denotes parabolic distance to the boundary.

The main theorem states that for any $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$0 there exists $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$1 such that if $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$2 — where $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$3 is the Lipschitz constant of the domain — then for every Neumann datum $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$4 the energy solution satisfies

$$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$5

with $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$6 the (averaged) nontangential maximal function. This is the first result on the Neumann problem in all dimensions for time-varying parabolic matrices under only this minimal smoothness hypothesis; prior work in the time-varying setting [AEN] imposed structural restrictions such as transversal independence of the coefficients.

Position within the boundary value program

The result completes, in the small-Carleson regime, the triad of parabolic boundary value problems for these operators. The Dirichlet problem was resolved for all $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$7 under the small Carleson condition in [DDH], building on [DH18]; the large-Carleson Dirichlet problem was settled in [DPP2]. The Regularity problem was recently solved in both regimes by the same authors in [DLP1], adapting ideas from the elliptic breakthroughs [DHP] and [MPT]. With the present paper, the parabolic theory for time-varying non-symmetric matrices is brought into full alignment with its elliptic counterpart as established in [DPR].

A structural obstruction distinguishes the parabolic case: the Regularity data involves a non-local half derivative in time, $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$8, alongside tangential spatial derivatives. Consequently, the elliptic roadmap of [DPR] — solve Regularity first, then transfer to Neumann via an $$\mathcal L u = -\partial_t u + \operatorname{div}(A\nabla u) = 0,$$9 equivalence of tangential and conormal derivatives on the boundary — cannot be followed directly. Instead, the authors derive nontangential estimates via the $\Omega = \mathcal O \times \mathbb R$0-adapted square function introduced in [DPP1], retaining from [DPR] the key observation that the conormal derivative combination $\Omega = \mathcal O \times \mathbb R$1 satisfies a useful PDE of its own.

Architecture of the proof

The proof on $\Omega = \mathcal O \times \mathbb R$2 proceeds through a perturbation reduction and a chain of square-function estimates. A smooth approximation lemma decomposes $\Omega = \mathcal O \times \mathbb R$3, where $\Omega = \mathcal O \times \mathbb R$4 has $\Omega = \mathcal O \times \mathbb R$5 entries, inherits ellipticity, and satisfies upgraded Carleson bounds including pointwise derivative estimates $\Omega = \mathcal O \times \mathbb R$6; a perturbation theorem of Ulmer [Ulm25] then transfers solvability from $\Omega = \mathcal O \times \mathbb R$7 back to $\Omega = \mathcal O \times \mathbb R$8. Fixing $\Omega = \mathcal O \times \mathbb R$9 with $\mathcal O$0 and writing $\mathcal O$1, it suffices to prove $\mathcal O$2 and interpolate against the adjoint Dirichlet problem $\mathcal O$3, which holds for all exponents by [DDH].

Three estimates form the core:

• $\mathcal O$4 estimates (Section 5). Via a stopping-time argument built on "enlarged" interior cones — defined as intersections of boundary cones, a construction that removes the cusp $\mathcal O$5 present in naively shifted cones — the authors prove good-$\mathcal O$6 inequalities yielding, for $\mathcal O$7 even, bounds of $\mathcal O$8, then of $\mathcal O$9 for tangential $A = [a_{ij}(X,t)]$0, and finally of $A = [a_{ij}(X,t)]$1, each by the corresponding $A = [a_{ij}(X,t)]$2-adapted square function plus small error terms involving $A = [a_{ij}(X,t)]$3 and $A = [a_{ij}(X,t)]$4.
• Bounds on the $A = [a_{ij}(X,t)]$5-adapted square functions (Section 6). Two lemmas control $A = [a_{ij}(X,t)]$6 and iterated integrals $A = [a_{ij}(X,t)]$7 in terms of the boundary $A = [a_{ij}(X,t)]$8 norm of $A = [a_{ij}(X,t)]$9 — which equals the Neumann datum — plus absorbable errors proportional to the small parameter $A$0. These are parabolic analogs of [DPR, Lemma 6.1], but the arguments differ substantially because one cannot invoke solvability of the Regularity problem without incurring the uncontrollable term $A$1.
• $A$2 for $A$3 (Section 7). A sawtooth-domain estimate with cutoff functions, again avoiding block-form assumptions on $A$4, closes the loop so that the $A$5 term can be absorbed.

Combining these gives $A$6 under an a priori finiteness assumption, with the small parameter absorbed on the right-hand side.

A priori finiteness and Moser iteration

Justifying the a priori bound $A$7 occupies considerable effort and, as the authors note, these estimates were absent from the literature even for smooth coefficients on unbounded domains. The argument approximates $A$8 by compactly supported smooth matrices $A$9, "ellipticizes" the problem by treating $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$0 as an extra spatial variable and solving an auxiliary elliptic Neumann problem via layer potentials, then reduces to an inhomogeneous problem with bounded, compactly supported right-hand side. A new bootstrap lemma — a parabolic energy estimate showing that solutions of $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$1 with data measured in $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$2 satisfy $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$3 — is applied alternately to tangential derivatives and to $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$4 (whose PDE follows from differentiating the equation for $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$5), iterating until $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$6; outside a large ball the heat kernel comparison principle supplies decay. This yields finiteness of $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$7 in every $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$8, $$d\mu = \sup_{B_{\delta(X,t)/2}}\left(\delta|\nabla A|^2 + \delta^3|\partial_t A|^2\right)dX\,dt$$9.

Bounded domains and localization

For bounded Lipschitz cylinders, solvability is bootstrapped from the unbounded case using three ingredients: the local estimate of [DLP2] for solutions with zero Neumann data on backward parabolic cubes; Brown's observation [B] that solutions with data localized in a short time interval enjoy bounds with absorbable small terms; and exponential decay of $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$0 away from the support of localized data, permitting summation over the space-time partition of unity. For unbounded domains above Lipschitz graphs, the Dahlberg–Kenig–Nečas–Stein pullback map, constant in $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$1, transfers the problem to the half-space without introducing drift terms, and the resulting matrix inherits the small Carleson condition.

Control of the half time derivative: a dichotomy

An appendix establishes a novel and somewhat surprising dichotomy concerning estimates of the form $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$2. On bounded domains no such estimate can hold: since Neumann data for these parabolic equations need not have mean zero (unlike the elliptic case, where data live modulo constants), the solution's time variation is unconstrained by $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$3. On unbounded domains above Lipschitz graphs, the estimate

$$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$4

does hold for $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$5. The authors state they expect the range $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$6 to hold as well but do not attempt to prove it here.

Limitations and open questions

The paper's scope is deliberately restricted to the small-parameter regime, and several questions remain open at the points where the results bear on them. First, solvability of the Neumann problem when the Carleson norm or the Lipschitz constant is large remains unresolved — even in the purely elliptic setting this is known only in dimension two, where the duality between Neumann and Regularity problems observed in [KR] applies; this duality does not persist in higher dimensions. Second, the extension of the half-time-derivative nontangential estimates to $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$7 on unbounded domains is conjectured but unproven. Third, the a priori finiteness machinery requires smoothness of coefficients and data before approximation; whether analogous estimates hold directly for rough coefficients is not addressed. Finally, the smallness constants depend on $$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$8, so uniform-in-$$d\mu = \delta^{-1}(\operatorname{osc}_B A)^2 dX\,dt$$9 solvability thresholds are not obtained.

Conclusion

This paper resolves the $\|\mu\|_C$0 Neumann problem, for all $\|\mu\|_C$1, for parabolic operators with non-symmetric, time-varying coefficients satisfying a small Carleson measure condition, on Lipschitz cylinders with sufficiently small Lipschitz constant. Together with [DDH] and [DLP1], it completes the small-Carleson theory of Dirichlet, Regularity, and Neumann problems for this class of operators, matching the elliptic state of the art while introducing techniques — enlarged interior cones, $\|\mu\|_C$2-adapted square function estimates for gradients and conormal derivatives, and a bounded/unbounded dichotomy for half-time-derivative control — that are specific to the parabolic setting. The large-Carleson regime remains the principal open problem, in both parabolic and higher-dimensional elliptic theories.