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Schauder estimates for parabolic $p$-Laplace systems

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Pith's one-line read Bounded weak solutions to parabolic p-Laplace systems with Hölder spatial coefficients have locally Hölder gradients for every exponent p>1.

desk verdict Genuinely new Schauder result, but the freezing argument rests on an unproved radius-removal claim in Proposition 5.1 that should be checked before accepting. read the letter →

arxiv 2507.15722 v1 pith:P7GZ5KZR submitted 2025-07-21 math.AP

Verena Bögelein , Frank Duzaar , Ugo Gianazza , Naian Liao , Christoph Scheven This is my paper · ORCID
classification math.AP MSC 35K6535K6735B4535B6535K9276S05
keywords Schauderestimatesparabolicp-LaplacesystemsgradientHölderregularityboundedweaksolutionsintrinsiccylindersCampanatodoublynonlinearequationsfastdiffusion
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 establishes gradient Schauder estimates for bounded weak solutions of the parabolic $p$-Laplace system with a coefficient that is only Hölder continuous in the space variable. The main theorem states that the spatial gradient $Du$ is locally Hölder continuous in space and time for every $p>1$, with explicit quantitative bounds: on a compact set $K$, the supremum of $|Du|$ is controlled by the oscillation of $u$, the distance to the parabolic boundary, and the regularization parameter $\mu$, and the Hölder modulus of $Du$ is given by a power of the intrinsic parabolic distance. This unifies gradient regularity theory that previously required the super-critical range $p>2N/(N+2)$, extends it to systems with coefficients, and covers the sub-critical range $1

What carries the argument

The central object is the intrinsic cylinder $Q_\rho^{(\lambda)}(z_o)=B_\rho(x_o)\times(t_o-\lambda^{2-p}\rho^2,t_o]$, whose time length is tuned to the gradient scale $\lambda$. The proof freezes the coefficient $a(x,t)$ to $a(x_o,t)$, compares $u$ to the solution $w$ of the frozen-coefficient Cauchy-Dirichlet problem on such a cylinder, and controls the comparison by Lemma 4.7's estimate $\int_{Q_R}|Du-Dw|^p\,dxdt\le CR^{\alpha_*p}\int_{Q_R}(\mu^2+|Du|^2)^{p/2}\,dxdt$. The decisive ingredient is the improved Campanato-type estimate (5.9), taken from [11], which gives decay of the mean oscillations of $Dw$ like $(\tau/R)^{\beta p}\lambda^p$ on intrinsic cylinders of fixed geometry. Combining this decay with the quantitative gradient bounds and an interpolation in an intermediate radius yields the Campanato estimate (5.21) for $Du$, from which the Lebesgue representative of $Du$ is shown to be Hölder continuous.

What would settle it

Take a frozen-coefficient system $\partial_t w-\operatorname{div}(b(t)(\mu^2+|Dw|^2)^{(p-2)/2}Dw)=0$ with $b$ depending only on time and compute the mean oscillations of $Dw$ over nested intrinsic cylinders $Q_\tau^{(\lambda)}\subset Q_R^{(\lambda)}$ with $\tau/R$ arbitrarily small, in dimensions $N\ge 2$ and exponents $p\le 2N/(N+2)$; a decay slower than $(\tau/R)^{\beta p}$ for some ratio outside the range permitted in [11] would falsify the radius-free version of Proposition 5.1 on which the main theorem rests.

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Extended reading notes

Core claim

In the paper's own terms, the central discovery is Theorem 1.1: for $p>1$, $\mu\in[0,1]$, and a coefficient $a$ satisfying (1.2), every bounded weak solution has $Du\in C^{\alpha_o,\alpha_o/2}_{\mathrm{loc}}(E_T,\mathbb{R}^{kN})$, with the quantitative local gradient bound (1.3) and the gradient Hölder estimate (1.4), where $\alpha_o$ and $C$ depend only on $N,p,C_0,C_1,\alpha,k$. The theorem is proved first under the a priori assumption that $Du$ is locally bounded and then by approximating the Hölder coefficient by smooth coefficients and passing to the limit, which also covers $\mu=0$. Below the critical exponent $p_*=2N/(N+2)$, the paper notes that weak solutions need not be locally bounded, citing [8] for counterexamples, so the boundedness hypothesis is essential, not merely technical. Above $p_*$, the method also yields gradient bounds depending only on $L^p$-integrals of $Du$. Theorem 6.1 then converts these estimates, via a time-insensitive Harnack inequality, into gradient and gradient-Hölder estimates for doubly nonlinear equations in the range $0<p-1<q<N(p-1)/(N-p)_+$, with explicit counterexamples showing that the endpoint cases $q=p-1$ and $q=N(p-1)/(N-p)_+$ are false.

Load-bearing premise

The load-bearing premise is that the improved Campanato estimate (5.9), imported without proof from [11] for coefficients depending only on time, remains valid on intrinsic cylinders of fixed geometry for all $p>1$ and all radius ratios once the radius restriction of [11] is dropped; if that imported estimate secretly requires a condition on $p$ or on the cylinder aspect ratio, Theorem 1.1 would lose its proof.

Editorial extensions

If this is right

  • Gradient Schauder estimates for parabolic $p$-Laplace systems now cover every $p>1$, with the boundedness assumption in the sub-critical range and no boundedness assumption above $2N/(N+2)$.
  • The quantitative estimates in terms of oscillation give explicit control of $\sup_K|Du|$ and of the Hölder modulus of $Du$ that is stable under approximation by smooth coefficients and persists in the limit $\mu\downarrow 0$.
  • For doubly nonlinear equations $\partial_t u^q-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0$ in the range $0<p-1<q<N(p-1)/(N-p)_+$, the paper obtains local Lipschitz and gradient-Hölder estimates at points where $u>0$, with constants depending only on $N,p,q$.
  • The same methods give quantitative extinction-time decay estimates for fast-diffusion solutions, including oscillation bounds for $u$ and $\nabla u$ near the extinction time.
  • The special case $p=2$ recovers the known gradient estimates for the singular porous medium equation in the range $(N-2)_+/N<m<1$.

Reading between the lines

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

  • If the theorem is correct, the freezing scheme should extend to boundary Schauder estimates and to coefficients that are Hölder in time, though the paper does not pursue those extensions.
  • The boundedness restriction below the critical exponent is probably intrinsic; a way to test this is to check whether the constants or the Hölder exponent in Theorem 1.1 must degenerate as $p$ approaches $1$ from above in a family of bounded approximating solutions.
  • The explicit extinction decay rates could be compared against the known self-similar solutions at the endpoint $q=N(p-1)/(N-p)_+$ to see whether the exponents in Corollary 6.7 are optimal.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript establishes local Hölder regularity of the spatial gradient for bounded weak solutions to the parabolic p-Laplace system ∂t u − div(a(x,t)(μ²+|Du|²)^{(p−2)/2}Du)=0 under the assumption that a is bounded, bounded away from zero, and Hölder continuous in the spatial variable, for every p>1 and μ∈[0,1]. The main theorem gives quantitative local L∞-gradient bounds and a local gradient Hölder estimate in terms of the oscillation of u, with constants depending only on N, p, C_o, C_1, α, and k. The proof combines self-contained energy estimates, Moser and De Giorgi iterations, comparison estimates for frozen-coefficient problems, and a Campanato-type freezing argument. As an application, the authors derive gradient estimates for a doubly nonlinear parabolic equation in a super-critical fast diffusion regime. The paper is transparent about the fact that in the sub-critical range 1<p≤2N/(N+2) the theorem is restricted to bounded solutions, which need not be locally bounded in general.

Significance. If the main theorem is correct, this is a substantial contribution: it provides a unified Schauder-type theory for parabolic p-Laplace systems with coefficients that are only Hölder continuous in space, covering all p>1, and it supplies explicit quantitative bounds rather than only qualitative regularity. The application to doubly nonlinear equations in Section 6 is nontrivial and gives concrete new estimates, including decay estimates at the extinction time. The paper is also unusually explicit about structural constants and about the limitations in the sub-critical range, and it contains long, detailed energy and comparison arguments that are valuable in themselves. However, the central claim currently rests on an imported radius-free Campanato estimate in Proposition 5.1 whose proof is only sketched by referring to the authors' earlier work [11]. The gap is localized, but it is load-bearing for the freezing argument and therefore for Theorem 1.1 and Section 6.

major comments (3)
  1. [§5.1, Proposition 5.1] The proof of Proposition 5.1 invokes [11, Theorem 1.3] after the rescaling v=A^{-1}w and then asserts that 'since the coefficients b(t) are independent of x, the upper bound ρ_o on the radius R can be avoided in the present situation.' No derivation is given for this assertion. The radius restriction in [11] may be tied to the parabolic sup-estimate and to the intrinsic coupling sup(|Dw|²+μ²)^{1/2}≤Aλ, neither of which disappears when the coefficient depends only on time. This is load-bearing because (5.9) is the only input for the Campanato estimates (5.17), (5.20), and (5.21), which in turn feed Proposition 5.3, Proposition 5.6, and Theorem 1.1. The statement of Proposition 5.1 also contains an impossible hypothesis: 'if Q^{(λ)}_{2R}(z_o) ⋐ Q^{(λ)}_R(z_o)' cannot hold for R>0, and the condition 'R∈(0,1/2 R)' is circular. The authors should either reproduce the relevant steps of [11, Section 5.2] in the time-only case, or state and prove the radius-free version as a self-contained lemma, and correct the cylinder inclusion in the statement.
  2. [§5.2, Proposition 5.2] In the sub-quadratic case 1<p<2, the proof of Proposition 5.2 applies Corollary 3.9 to w with the parameter λ. Corollary 3.9 is stated under the assumption λ≥μ, but assumption (5.3) alone yields only λ≥μ/A, with A not necessarily equal to 1. The sentence in the proof that says the application is justified 'since λ≥μ/A as a consequence of (5.3)' does not meet the hypothesis of Corollary 3.9 as stated. If A>1, either Corollary 3.9 must be reformulated to allow λ≥μ/A with constants depending on A, or the argument must first rescale the solution. This matters because (5.14) is used to derive the intrinsic coupling (5.16), which in turn is needed to apply Proposition 5.1 and obtain (5.17). This gap is technical and likely fixable, but it must be closed before the estimates in Section 5.2 are fully justified.
  3. [§5.5, Proof of Theorem 1.1] The approximation argument in the proof of Theorem 1.1 relies on applying the a priori estimates from Sections 5.3 and 5.4 to the regularized solutions u_i. For this to be valid, the constant A in (5.3) must be controlled uniformly in i; the text states that the constants in Corollary 5.4 and Proposition 5.6 are independent of i, which is correct because of (5.35). It would be helpful to state this uniformity explicitly at the point where A is introduced in the proof of Theorem 1.1, since the same A enters the exponent β in Proposition 5.1 and hence the final Hölder exponent α_o. As written, the reader has to assemble this from (5.35)-(5.38); the argument is sound but the presentation would be clearer if this uniformity were highlighted.
minor comments (4)
  1. [Title and headings] There are typographical errors in the running text and headers, e.g. 'SCHAUDER ESTIMA TES' and 'A PPLICATIONS'. These should be corrected.
  2. [§3.4, Proposition 3.8] The proof uses 'since N≥2' to ensure that the chosen m satisfies m>3, which is needed for the quantitative estimate in Lemma 3.3. The proposition and the rest of the paper do not exclude N=1. Either state N≥2 as an assumption in this proposition or modify the choice of m so that the argument covers N=1 as well.
  3. [§6.5] In the endpoint discussion, the expression q=N(p-1)/(N-p)_+ is undefined when N≤p. The convention for (N-p)_+ in the endpoint case should be made explicit, and the counterexample should be stated only in the range where N>p.
  4. [§5.1, Proposition 5.1] The notation in the hypothesis 'R∈(0,1/2 R)' is circular and appears to be a typo; the intended condition is presumably R≤ρ_o/2 or a similar bound involving an auxiliary radius. Please rewrite the statement with unambiguous notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the Schauder theorem is derived from independent energy, comparison, and approximation arguments; the load-bearing same-author citations are independent support, with one radius-extension assertion that is a proof gap, not a circularity.

full rationale

No circularity is present in the derivation chain. Sections 3 and 4 are self-contained: the gradient bounds are obtained from energy estimates, Moser/De Giorgi iterations, and comparison estimates whose constants are resolved by standard re-absorption and Young inequalities, not fitted to the conclusion. Section 5 freezes coefficients and invokes Proposition 5.1, which is imported from the authors' published paper [11, Theorem 1.3]; that theorem concerns coefficients depending only on time and does not assume the target x-Hölder Schauder estimate, so it is independent support rather than a circular input. The only substantive concern is the sentence in the proof of Proposition 5.1: 'Since the coefficients b(t) are independent of x, the upper bound ρ_o on the radius R can be avoided in the present situation.' This is an unproved extension of [11] and is load-bearing for the full range p>1, but it is a proof-completeness and correctness risk, not an equation-level circularity: no quantity in Theorem 1.1 is defined in terms of the estimate it predicts, and no fitted parameter is renamed as a prediction. The approximation argument in Section 5.5 and the application in Section 6 similarly use standard mollification, L^p convergence, and Harnack inputs from [8] whose assumptions do not contain the Schauder result.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted: every auxiliary exponent (α_o, β, κ, δ, θ, θ_2 through θ_5, ν, d_1, d_2) is an explicit algebraic function of the structural data N, p, α, C_o, C_1, k, chosen to close iteration estimates. The central claim rests on the stated hypotheses (boundedness of u in the sub-critical range, coefficient condition (1.2)), on standard machinery from DiBenedetto's monograph and classical papers, and on two author-overlapping black boxes: the Campanato estimate of [11] (published, imported as Proposition 5.1) and the boundedness/Harnack results of [8] (companion preprint, imported in Section 6.6). No invented entities are introduced; the auxiliary objects (comparison solution w, v = u^q, Lebesgue representative Γ_zo) are standard constructions.

assumptions (6)
  • domain assumption Weak solutions are assumed bounded in the sub-critical range 1 < p ≤ 2N/(N+2) (hypothesis of Theorem 1.1)
    Load-bearing: the paper states (Section 1) that below p* solutions need not be locally bounded (counterexamples credited to [8]), so without this hypothesis the theorem is false in that range.
  • domain assumption Coefficient condition (1.2): C_o ≤ a ≤ C_1 and |a(x,t) − a(y,t)| ≤ C_1 |x − y|^α
    The central claim is conditional on this structural hypothesis stated in Section 1.
  • domain assumption Campanato estimate for frozen time-dependent coefficients, [11, Theorem 1.3], imported as Proposition 5.1
    Used without proof in Section 5.1; the paper asserts the radius restriction of [11] is removable when b(t) is independent of x, which is load-bearing for fixed-geometry intrinsic cylinders.
  • domain assumption Boundedness and time-insensitive Harnack inequality for the doubly nonlinear equation, [8, Theorems 1.5 and 1.11]
    Invoked in the proof of Theorem 6.1 (Section 6.6); [8] is an unreviewed companion preprint by the same five authors, and the manuscript states part of it was extracted from [8].
  • standard math DiBenedetto-Friedman gradient Hölder theory and DiBenedetto's monograph [17] (Sobolev embedding, iteration lemmas, Chapter VIII and IX machinery)
    Background tools used throughout Sections 3 and 5; classical published results.
  • standard math Acerbi-Fusco and Giaquinta-Modica monotonicity estimates (Lemma 2.3), Steklov averaging, standard inequalities
    Standard analytical tools collected in Section 2.

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Pith. "Pith review of Schauder estimates for parabolic $p$-Laplace systems." pith.science (2026). https://pith.science/paper/P7GZ5KZR

@misc{pith2026250715722,
  author       = {Pith},
  title        = {Pith review of: Schauder estimates for parabolic $p$-Laplace systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P7GZ5KZR}},
  note         = {Machine review of arXiv:2507.15722}
}
abstract

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big( a(x,t)\big(\mu^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $\mu\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and is H\"older continuous in the space variable $x$. As an application, we prove H\"older estimates for the gradient of weak solutions to a doubly non-linear parabolic equation in the super-critical fast diffusion regime.

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