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Boundary H\"older gradient estimates for parabolic $p$-Laplace type equations

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Pith's one-line read Viscosity solutions of parabolic p-Laplace type equations gain pointwise $C^{1,\alpha}$ regularity at the boundary when data and boundary are $C^{1,\alpha}$.

desk verdict A credible boundary C^{1,α} regularity paper with a real gap in the written proof: the ε→0 approximation in Theorem 6.4 is deferred, and a referee should demand it. read the letter →

arxiv 2506.01018 v1 pith:EC6HXQMD submitted 2025-06-01 math.AP

Se-Chan Lee , Yuanyuan Lian , Hyungsung Yun , Kai Zhang This is my paper · ORCID
classification math.AP MSC 35B6535D4035K5535K6535K6735K92
keywords boundaryregularityparabolicp-LaplaceequationviscositysolutionsC^{1\alpha}estimatesdegenerateequationssingularpointwiseelliptic
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

This paper establishes the first boundary pointwise H\"older gradient ($C^{1,\alpha}$) regularity for viscosity solutions of parabolic $p$-Laplace type equations with general Dirichlet data. The main theorem says that if the boundary data $g$ and the parabolic boundary $\partial_p\Omega$ are $C^{1,\alpha}$ at a boundary point, then the solution is well approximated there by a time-independent linear polynomial, with an error of order $|x|^{1+\alpha}$ in space and $|t|^{(1+\alpha)/(2-\alpha\gamma)}$ in time. The result covers the full range $-1<\gamma<\infty$, $12$), singular ($1

What carries the argument

The load-bearing object is the family of regularized operators $$\mathcal{P}^\varepsilon_{a,\nu}u = u_t - (|\nu Du+a|^2+\$varepsilon^{2}$)^{\gamma/2}\left(\delta_{ij}+(p-2)\frac{(\nu u_i+a_i)(\nu u_j+a_j)}{|\nu Du+a|^2+\$varepsilon^{2}$}\right)u_{ij},$$ in which the degenerate factor $|Du|^\gamma$ is replaced by a smoothed version with parameters $a\in\mathbb R^n$ and $\nu\in[0,1]$. The proof splits every situation into a nondegenerate regime ($|a|\gg\nu$, where uniformly parabolic barriers, strong maximum principle, Harnack inequality, and Hopf lemma apply) and a degenerate regime ($|a|\ll\nu$, where a two-parameter intrinsic scaling $y=x/r$, $s=t/(\rho^{-\gamma}r^2)$, $v=u/(r\rho)$ and a decay lemma for $|Du|$ are used). These regimes are combined through an iteration for the model problem with flat boundary and zero data, followed by small-perturbation and compactness arguments that reach general boundaries and nonzero Dirichlet data.

What would settle it

Find a radially symmetric viscosity solution of the elliptic $p$-Laplace equation on a ball with $C^{1,\alpha}$ boundary data whose boundary error grows faster than $C|x|^{1+\alpha}$ at a boundary point; such an example would falsify Corollary 1.13 and hence Theorem 1.7.

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Extended reading notes

Core claim

The central claim, Theorem 1.7, is that a viscosity solution $u$ of $\mathcal{P}u=f$ in $\Omega\cap Q_1$ with $u=g$ on $\partial_p\Omega\cap Q_1$ belongs to $C^{1,\alpha}(0,0)$ when $\gamma>0$, and to $C^{1,\alpha}_\gamma(0,0)$ when $\gamma\le 0$, provided that $g$ and the boundary $\partial_p\Omega\cap Q_1$ are $C^{1,\alpha}_\gamma$ at $(0,0)$ for some $\alpha\in(0,\bar\alpha)$. Concretely, there is a time-independent linear polynomial $L$ with $|u(x,t)-L(x)|\le C(|x|^{1+\alpha}+|t|^{(1+\alpha)/(2-\alpha\gamma)})$ in $\Omega\cap Q_1$, where $\bar\alpha<\min\{1/2,1/2(1+\gamma)\}$ is universal. The proof is purely non-variational: it avoids integral estimates and boundary flattening, instead proving the estimate first for a model problem with flat boundary and zero data, then transferring it to curved boundaries and general data through perturbation and compactness arguments. A direct corollary, Corollary 1.13, gives the analogous boundary pointwise $C^{1,\alpha}$ regularity for elliptic $p$-Laplace equations.

Load-bearing premise

The proof depends on the assumption that the uniform boundary estimates proved for the $\varepsilon$-regularized smooth model problem remain valid in the limit $\varepsilon\to0$, so that they transfer from approximating solutions to the viscosity solution of the original degenerate equation.

Editorial extensions

If this is right

  • If Theorem 1.7 is correct, viscosity solutions of (1.1) enjoy boundary pointwise $C^{1,\alpha}$ regularity at every boundary point where $g$ and $\partial_p\Omega$ are $C^{1,\alpha}_\gamma$, and combining with interior estimates gives global $C^{1,\alpha}$ regularity as stated in Theorem 1.17.
  • The elliptic corollary, Corollary 1.13, provides the first boundary pointwise $C^{1,\alpha}$ regularity for viscosity and weak solutions of the classical elliptic $p$-Laplace equation with $C^{1,\alpha}$ boundary data and domain.
  • When $\gamma=0$ and $p$ is close to $2$, the universal exponent $\bar\alpha$ can be chosen close to $1$, so the boundary estimate gives a near-optimal H\"older exponent for the gradient.
  • The authors state that the non-variational technique is flexible enough to be applied to more complicated problems, including fully nonlinear degenerate and singular parabolic equations.

Reading between the lines

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

  • The time exponent $(1+\alpha)/(2-\alpha\gamma)$ in (1.8) comes from the intrinsic two-parameter scaling; one could test whether it is sharp for $\gamma\neq 0$ by studying explicit self-similar solutions with power-like boundary behaviour.
  • If the $\varepsilon\to0$ stability step can be made fully quantitative, the same perturbation-and-compactness approach should transfer to fully nonlinear degenerate and singular parabolic equations, where interior estimates are already available.
  • The elliptic Corollary 1.13 suggests that boundary pointwise $C^{1,\alpha}$ regularity holds for the $p$-Laplacian in arbitrary dimension with $C^{1,\alpha}$ data; the main thing to verify independently would be the uniform stability of the smooth approximations at the boundary.
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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 / 5 minor

Summary. The paper develops boundary regularity theory for viscosity solutions of parabolic p-Laplace type equations of the form u_t - |Du|^γ (δ_ij + (p-2) u_i u_j / |Du|^2) u_ij = f. The main result, Theorem 1.7, asserts a pointwise boundary C^{1,α} estimate at (0,0) under C^{1,α}_γ assumptions on the boundary data and the lateral boundary, with a universal exponent ᾱ; a global version is given in Theorem 1.17, and a new elliptic boundary pointwise C^{1,α} result for the p-Laplacian is deduced in Corollary 1.13. The proof proceeds through a non-variational strategy: interior C^{0,1} and C^{1,α} estimates, boundary estimates for a flat model problem via barrier constructions and the two-parameter intrinsic scaling, a degeneration/nondegeneration dichotomy following Imbert-Jin-Silvestre, and finally a compactness-based perturbation argument for curved boundaries and nonzero data.

Significance. If the main theorem is correct, it provides the first boundary pointwise C^{1,α} regularity for this family of singular/degenerate parabolic equations in the viscosity setting, covering both the degenerate range γ>0 and the singular range γ≤0; the elliptic corollary for the p-Laplacian is also new. The proof is genuinely non-variational and introduces a systematic combination of two-parameter scaling, barrier methods, and small-perturbation compactness arguments, which is likely to be influential for further boundary regularity results. The paper is careful with the structure of the generalized operator P^ε_{a,ν} and with the distinction between degenerate and nondegenerate regimes. However, the current text contains several explicitly omitted proofs at load-bearing points, so the result is not fully verifiable as submitted.

major comments (3)
  1. [Section 5, Lemma 5.7] Lemma 5.7 is stated without proof: 'Since its proof is exactly the same as that of [22, Corollary 4.2], we omit it.' This lemma is load-bearing: it produces the decay chain (6.3)-(6.4) used in every case of Theorem 6.1, and its conditions (5.13)-(5.15) involve both ε and |a| in a way that is specific to the boundary model problem. The boundary setting is not identical to the interior Corollary 4.2 of [22], because the cylinders are half-balls Q^{ρ+}_r and the scaling is the two-parameter family. A reader cannot verify the iteration constants or the stopping-time argument from the paper alone. Please supply the proof or a complete reduction to [22, Corollary 4.2] with all rescalings and boundary conditions written out.
  2. [Section 6, Theorem 6.4] Theorem 6.4 passes from smooth solutions of the regularized equation P^ε_a u=0 to viscosity solutions of P_a u=0 by the sentence 'By an approximation (see [22, Section 5])'. This is the critical interface between the uniform estimates of Section 5 and the model problem used in Lemma 7.2 and thereafter. To be rigorous, the proof must show: (i) for each ε>0 there exist smooth solutions u^ε of P^ε_a u^ε=0 in Q_1^+ with u^ε=0 on S_1 approximating the given viscosity solution; (ii) the constants in Theorem 6.3, especially C in (6.12)-(6.13), are independent of ε and of the approximating data; and (iii) a subsequence u^ε converges locally uniformly to the given u, so the limit is the same solution, not another one. None of these is written out, and the cited [22, Section 5] is an interior argument that does not by itself control the boundary condition u=0 on S_1. Since Theorem 6.4 is invoked in every later perturbation step, the main theorem is not fully established unless this approximation step is supplied.
  3. [Section 7, Lemma 7.6] Lemma 7.6 is the γ≤0 counterpart of Theorem 7.4, and its proof is omitted with the comment that it is similar. This is not a redundant repetition: for γ≤0 the conclusion is u∈C^{1,α}_γ(0,0) with time exponent (1+α)/(2−αγ), the scalings in Lemmas 7.3 and Theorem 7.4 use t ~ r^{2−αγ}, and the condition α(1+γ)≤1 plays a different role. Since Theorem 1.7 for γ≤0 rests directly on Lemma 7.6, the omission should be filled at least by an explicit outline of the modified iteration, the changed scaling exponents, and the points where the proof of Theorem 7.4 has to be adjusted.
minor comments (5)
  1. [Throughout] Displayed results are labeled as Lemmas but the text repeatedly refers to them as Theorems (for example 'Theorem 2.4', 'Theorem 2.6', 'Theorem 2.8', 'Theorem 3.1', 'Theorem 4.6', 'Theorem 5.1', 'Theorem 6.1', 'Theorem 7.2'). Please unify the numbering and cross-references.
  2. [Lemma 4.6] In the proof, the displayed expression 'P^ε_{a,ν} <0' is missing the function v; it should read 'P^ε_{a,ν} v <0'.
  3. [Remarks 2.2 and 2.7] Remark 2.2 refers to 'Theorem 2.1' and Remark 2.7 refers to 'Theorem 2.6', but the objects in question are Definition 2.1 and Lemma 2.6. Similar mismatches occur in Remarks 1.2 and 1.6.
  4. [Proof of Lemma 7.4] After the rescaling, the assertion '∥(∂_p \tilde Ω)_1∥_{C^{1,α}(0,0)} ≤ θ' is made without derivation. Since the hypotheses of Lemma 7.4 give scale-invariant oscillation bounds rather than an explicit C^{1,α} norm, a one-line justification of this implication would remove ambiguity.
  5. [Statement of Lemma 7.4] The phrase 'where we choose β=α and ∥(∂_pΩ)_1∥_{C^{1,β}(0,0)}≤1' is ambiguous, because the norm is not listed among the displayed assumptions; clarify that this is a normalization condition and explain how it is achieved before applying Theorem A.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the boundary C^{1,α} estimate is derived from an independently established model problem and external small-perturbation theorems.

full rationale

The derivation chain is a standard top-down reduction: the main theorem (Theorem 1.7) is proved from the flat-boundary model problem (Theorem 6.4) via compactness and perturbation arguments (Lemma 7.2, Theorem 7.4), and Theorem 6.4 is itself obtained from smooth solutions of the regularized equation P^ε_a, with the passage to viscosity solutions justified by an approximation argument attributed to the external paper [22, Section 5]. No parameter is fitted from the target estimate, no conclusion is assumed in its own proof, and no load-bearing argument reduces to a self-citation. The self-citations that occur are to definitions, context, and remarks: Definition 1.5 adapts the boundary regularity class from [37, Definition 1.4], Remark 1.9 mentions [34, Theorem 1.4], and the interior fully nonlinear result [32] is cited only for comparison. The main interior C^{1,α} regularity is quoted from external works [2,4,5,22], and the small perturbation lemmas in Appendix A are proved in the paper rather than merely imported. The omitted proof of Lemma 5.7 and the terse approximation step in Theorem 6.4 are potential completeness or correctness risks, but they are not circular reductions: they concern the existence and stability of smooth approximating solutions for a degenerate equation, which is a distinct premise from the boundary regularity claim itself. The definition of the pointwise class C^{1,α}_γ is tailored to the intrinsic scaling of the equation, but the theorem still proves a nontrivial estimate for solutions and does not obtain it by definition. Overall, no circular step is identifiable in the text.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is pure mathematics with no data fitting; the central claim is supported by an analytic proof. The only 'free parameters' are universal constants chosen during proofs, which are not free in the empirical sense. The main external assumptions are the interior regularity theorems for the unit-ν operator and the classical theory of uniformly parabolic equations.

assumptions (4)
  • standard math Stability and comparison principles for viscosity solutions of P^ε_{a,ν}
    Used throughout; based on [45, Theorem 6.1], [10, Proposition 3], and Remark 2.3.
  • domain assumption Interior C^{1,α}_γ regularity for the operator with ν=1 (i.e., for P^ε_{0,1}) with nonhomogeneous f
    Borrowed from [2, Theorem 1.1], [4, Theorem 1.1], [5, Theorem 1.1], [22, Theorem 1.1]; used in Theorem 2.10, Sections 6 and 7.
  • standard math Classical Schauder and Harnack estimates for linear uniformly parabolic equations with constant coefficients
    Used in the proofs of Lemma 4.4, Lemma 4.6, and Appendix A (limit equation A.7).
  • domain assumption Equivalence of viscosity and weak solutions for the elliptic p-Laplace equation (for Corollary 1.13)
    Cited from [27,28,42] and used to transfer the parabolic result to the elliptic setting.

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Pith. "Pith review of Boundary H\"older gradient estimates for parabolic $p$-Laplace type equations." pith.science (2026). https://pith.science/paper/EC6HXQMD

@misc{pith2026250601018,
  author       = {Pith},
  title        = {Pith review of: Boundary H\"older gradient estimates for parabolic $p$-Laplace type equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EC6HXQMD}},
  note         = {Machine review of arXiv:2506.01018}
}
abstract

In this paper, we study the boundary regularity for viscosity solutions of parabolic $p$-Laplace type equations. In particular, we obtain the boundary pointwise $C^{1,\alpha}$ regularity and global $C^{1,\alpha}$ regularity.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Harnack inequality for degenerate fully nonlinear parabolic equations

    math.AP 2025-06 accept novelty 8.0 of 10

    An intrinsic Harnack inequality, with two distinct waiting times, is proven for nonnegative viscosity solutions of degenerate fully nonlinear parabolic equations, yielding local Holder continuity.

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