Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality

Hongjie Dong; Peng Fa; Yi Ru-Ya Zhang; Yuan Zhou

REVIEW 1 major objections 5 minor 36 references

Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A dimension-weighted Hessian inequality yields second-order Sobolev regularity for p-Laplacian equations and fully settles the planar parabolic normalized case.

desk verdict A solid, self-contained regularity paper built on a new algebraic inequality; the main theorems are new and the proofs are mostly careful, with one standard-but-unproven approximation step that a referee should ask about. read the letter →

arxiv 1908.01547 v2 pith:J34AEHOI submitted 2019-08-05 math.AP

Hongjie Dong , Peng Fa , Yi Ru-Ya Zhang , Yuan Zhou This is my paper · ORCID

classification math.AP MSC 35B6535J9235K9235D40

keywords p-Laplaciannormalizedp-Laplaceequationinfinity-Laplaciansecond-orderregularitySobolevviscositysolutionsparabolicequationsHessianestimates

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 a pointwise structural inequality for smooth functions in $n$ dimensions: the algebraic interaction between the Hessian, the Laplacian $\Delta v$, and the infinity-Laplacian $\Delta_\infty v=D^2vDv\cdot Dv$ is controlled by a dimension-dependent error term, and the control is an exact identity in the plane. From this single inequality it derives second-order Sobolev regularity for solutions of the $p$-Laplace equation, the parabolic normalized $p$-Laplace equation, and the degenerate parabolic $p$-Laplace equation. In two dimensions it completely answers an open question by showing that viscosity solutions of the parabolic normalized $p$-Laplace equation have spatial Hessian and time derivative in $L^q_{\rm loc}$ for some $q>2$ for every $p\ne2$. Because the argument avoids convexity or monotonicity of the $p$-Laplacian, the same inequality governs all three equation classes and yields explicit estimates that remain uniform as $p$ approaches $2$.

What carries the argument

The load-bearing object is the dimension-weighted pointwise inequality (2.1), which bounds the algebraic expression $|D^2vDv|^2-\Delta v\,\Delta_\infty v-\tfrac12(|D^2v|^2-(\Delta v)^2)|Dv|^2$ by the nonnegative quantity $|D^2v|^2|Dv|^2-|D^2vDv|^2$ times $(n-2)/2$. Its proof diagonalizes the Hessian at a point, reducing the inequality to an $n$-term eigenvalue statement whose direction is the unit vector $Dv/|Dv|$, and then invokes an elementary vector inequality. The argument applies this inequality to smooth regularized solutions of the elliptic $p$-Laplace equation and of the two parabolic equations under study; it yields uniform-in-$\varepsilon$ estimates on $|D^2u_\varepsilon|^2$, $|u_{\varepsilon,t}|^2$, and weighted products of Hessian and gradient. Those uniform estimates are what pass to the limiting solution through compactness and a higher-integrability step.

What would settle it

Take an explicit planar solution of the parabolic normalized $p$-Laplace equation, for instance a radial or self-similar profile, and compute whether $\int_{Q_r}(|D^2u|^q+|u_t|^q)\,dx\,dt$ is finite for some $q>2$; a single example where the integral diverges for every $q>2$ would refute Theorem 1.3. Similarly, a $p$-harmonic function for which $|Du|^{(p-\gamma)/2}Du$ fails to lie in $W^{1,2}_{\rm loc}$ for some $\gamma<\gamma_{n,p}$ would refute Theorem 1.1.

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Extended reading notes

Core claim

The paper's central claim is that the planar identity $$ |$D^{2}$vDv|^2-\$\Delta$ v\,\Delta_\infty v=\tfrac12(|$D^{2}$v|^2-(\$\Delta$ v)^2)|Dv|^2 $$ admits a higher-dimensional replacement: for every smooth $v$, $$ \Bigl||$D^{2}$vDv|^2-\$\Delta$ v\,\Delta_\infty v-\tfrac12(|$D^{2}$v|^2-(\$\Delta$ v)^2)|Dv|^2\Bigr| \le \frac{n-2}{2}\bigl(|$D^{2}$v|^2|Dv|^2-|$D^{2}$vDv|^2\bigr), $$ where $\Delta_\infty v=D^2vDv\cdot Dv$. Using this inequality on regularized solutions, the paper proves that $p$-harmonic functions satisfy $|Du|^{(p-\gamma)/2}Du\in W^{1,2}_{\rm loc}$ for $\gamma<\min\{p+\frac{n}{n-1},\,3+\frac{p-1}{n-1}\}$; that viscosity solutions of the parabolic normalized $p$-Laplace equation have $D^2u,u_t\in L^q_{\rm loc}$ for some $q>2$ in the exponent range $(1,2)\cup(2,3+\frac{2}{n-2})$, which for $n=2$ means every $p\ne2$; and that for the degenerate parabolic $p$-Laplace equation, $D^2u\in L^2_{\rm loc}$ and $u_t\in L^2_{\rm loc}$ for $1<p<3$, with the upper endpoint sharp.

Load-bearing premise

The proof works first on smooth approximate solutions; if those approximations did not have uniformly bounded gradients and did not converge to the true solution, the second-order estimates would not carry over to the limit.

Editorial extensions

If this is right

The exponent range for the weighted gradient quantity in Theorem 1.1 improves the earlier bound $\gamma\le2$ for every $p\ne2$ and every dimension $n\ge2$.
For the parabolic normalized $p$-Laplace equation in the plane, the open question on second-order regularity is settled for all $p\ne2$, with the stronger conclusion that the integrability exponent is $q>2$ rather than merely $2$.
For $n\ge3$, viscosity solutions of the parabolic normalized equation have $D^2u$ and $u_t$ in $L^q_{\rm loc}$ for some $q>2$ throughout $p\in(1,2)\cup(2,3+\frac{2}{n-2})$, a wider range than the coefficient-degeneracy approach could reach.
For the degenerate parabolic $p$-Laplace equation, the range $p\in(1,3)$ is sharp for spatial $W^{2,2}$-regularity: an explicit solution has $|D^2w|$ comparable to $|x_1|^{(2-p)/(p-1)}$, which is in $L^2_{\rm loc}$ exactly when $p<3$.
All estimates come with constants that do not blow up as $p\to2$, so the results connect continuously to the classical theory at $p=2$.

Reading between the lines

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

Because the fundamental inequality is purely algebraic and needs no convexity or monotonicity of the $p$-Laplacian, it may transfer to other equations whose operators interpolate between Laplacian and infinity-Laplacian, such as game-theoretic or image-processing parabolic models; the paper does not explore these applications.
One way to probe sharpness is to compute the weighted gradient quantity for explicit power-type $p$-harmonic functions and check whether $W^{1,2}$ integrability fails as $\gamma$ approaches the paper's upper bound; the paper does not carry out this endpoint test.
The authors conjecture that for $n\ge3$ and $p\ge3+\frac{2}{n-2}$, the expression $|D^2u|^2-(\Delta u)^2$ can change sign for some $p$-harmonic function; if verified, the exponent ranges in the second-order results would be sharp, and the method's limitation would be intrinsic rather than technical.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read this one. It is a genuinely strong regularity paper. The central object is the pointwise inequality (1.2), a dimension-dependent bound on |D^2v Dv|^2 - Δv Δ_∞v - 1/2[|D^2v|^2 - (Δv)^2]|Dv|^2. The proof via eigenvalues and Cauchy-Schwarz is short, self-contained, and correct. Everything else hangs off it, and the paper is honest about what is new and what is reproof. Theorem 1.3 answers the Høeg-Lindqvist question for n=2 with higher integrability q>2; Theorem 1.5 gives W^{2,2} for the parabolic p-Laplacian with sharp range p<3, with a clean sharpness example. Those are real advances. The proofs are long and technical, but the structure is clear: Lemma 2.1, pointwise estimates for ε-regularized solutions, uniform L^2 bounds on D^2u_ε and u_ε,t, then Gehring/compactness to pass to the limit.

The soft spot, as flagged, is the approximation step. The paper cites [22] and [11,36] for convergence and uniform gradient bounds of the regularized problems, but those references treat the limiting degenerate equations, not the ε-regularized ones with rough boundary data. This is a genuine gap, though a standard one; I expect it can be filled by combining the cited interior estimates with standard boundary arguments. Similarly, in Section 5 the uniform L∞ bound on Du_ε is asserted without a precise citation. The boundary terms at t=0 in Lemmas 4.7 and 4.8 are controlled only if Du_ε is bounded uniformly up to the initial slice; the paper does not discuss this. These are not fatal, but a referee should ask for precise statements.

Corollary 1.2 partly reproves known Cordes-condition results, and the paper says so. That is fine.

Overall: the main theorem is new, the proof is mostly self-contained, and the cited background is standard. I would send it to a serious referee and would cite it if I worked in this area.

Referee Report

1 major / 5 minor

Summary. The paper proves a pointwise algebraic inequality (Lemma 2.1, Eq. (1.2)) controlling the structure of ΔvΔ∞v in terms of |D^2v| and Dv, and then uses this inequality as the central tool to establish second-order Sobolev regularity for p-harmonic functions (Theorem 1.1, Corollary 1.2), for viscosity solutions of the parabolic normalized p-Laplace equation (Theorem 1.3), and for weak/viscosity solutions of the parabolic p-Laplace equation (Theorem 1.5). The results include improved ranges of γ and p, a complete answer for n=2 to an open question of Høeg and Lindqvist, and a sharpness example for the range p<3 in Theorem 1.5. The proofs pass through ε-regularized smooth solutions and then take limits using known convergence and uniform gradient bounds.

Significance. If the claims hold, this is a significant contribution to the regularity theory of degenerate elliptic and parabolic equations. The fundamental inequality (1.2) is new, is proved from scratch via an elementary spectral decomposition, and is likely to be a useful tool beyond the applications considered here. The paper also gives explicit quantitative estimates and a clean sharpness calculation for the parabolic p-Laplace equation. The central analytic arguments are self-contained apart from standard approximation facts, and the main results improve previously known ranges in the elliptic case and are the first higher-integrability results in the parabolic normalized case for n=2.

major comments (1)

[Sections 4 and 5, Eqs. (1.12) and (1.13)] The passage from the regularized solutions u_ε to the limit solution u relies on the assertions that u_ε ∈ C^∞(U_T) ∩ C^0(overline{U_T}), Du_ε ∈ L^∞(U_T) uniformly in ε, and u_ε → u in C^0(U_T). In Section 4 this is attributed to [22], and in Section 5 to [11,36]. However, the cited papers concern the unregularized degenerate equations, not the ε-regularized equations (1.12) and (1.13) with boundary data inherited from a viscosity solution. These approximation properties are load-bearing: without them the weak-limit identification of D^2u_ε and u_{ε,t} in Theorems 1.3 and 1.5 collapses. The authors should either provide precise statements or proofs from the standard theory of uniformly parabolic quasilinear equations, or give correct references that cover the regularized problems uniformly in ε. This gap is likely fillable but must be addressed in the manuscript.

minor comments (5)

[Theorem 1.3 and Corollary 1.2] The range p ∈ (1,2) ∪ (2, 3+2/(n−2)) is stated without comment for n=2, where the expression 3+2/(n−2) is undefined; it should be interpreted as the whole interval (1,∞), and this convention should be stated explicitly.
[Theorem 1.3, Eq. (1.7)] The statement 'u_t, D^2u ∈ L^q_loc(Ω)' should read 'u_t, D^2u ∈ L^q_loc(Ω_T)', since the estimates are on space-time cylinders Q_r ⊂ Ω_T.
[Section 4, Lemma 4.1 and proof of Theorem 1.3] The notation Q_r and Q(0,r) is used inconsistently: the definition Q_r(z,s) = (s−r^2,s) × B(z,r) gives a cylinder ending at time s, but the proof of Lemma 4.1 appears to translate time so that the bottom of the cylinder is t=0. Please clarify the time normalization and the appearance of integrals over B_{2r} at t=0 in Lemmas 4.7 and 4.8.
[Lemma 4.6 and Lemma 4.7] The boundary terms involving ln[|Du_ε(x,0)|^2+ε] are controlled only after invoking uniform boundedness of Du_ε and the prefactor ε; this should be stated explicitly, since ln[|Du_ε|^2+ε] is not bounded below uniformly as ε→0.
[Abstract and Section 1.1] In the abstract, the formula for γn,p appears as min{p + (n−1)/n, 3+(p−1)/(n−1)}, while Theorem 1.1 states min{p + n/(n−1), 3+(p−1)/(n−1)}. The theorem version is the correct one; the abstract should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the fundamental inequality is proved from scratch and the self-citation [24] is only motivational.

full rationale

The central derivation is self-contained. Lemma 2.1, the fundamental inequality, is proved directly from linear algebra (Lemma 2.2) via spectral decomposition, with no use of the planar identity (1.1) except as background motivation attributed to [24]. When n=2, inequality (2.1) reproduces (1.1), but the proof does not rely on [24]; the identity is a special case of the paper's own Lemma 2.1. The main theorems follow from (2.1), Lemma 2.3, elementary integration by parts, and standard analytic tools such as Gehring's lemma, Sobolev-Poincare inequalities, and compact embedding. The approximation and convergence facts cited from [33, 26, 10, 22, 11, 36] are external regularity results, not results of this paper, and no fitted parameter is used to produce the predicted second-order estimates. The only self-citation, [24] by Koch-Zhang-Zhou, is not load-bearing: it is mentioned to motivate the planar identity but is not needed to prove Theorem 1.1, Theorem 1.3, or Theorem 1.5. Concerns about whether the cited approximation theory applies verbatim to the epsilon-regularized problems are correctness or technical-gap concerns, not circularity, because the paper does not define its outputs in terms of those citations or rename fitted quantities as predictions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proofs rely on standard elliptic and parabolic regularity theory, Gehring's lemma, and compact embedding results, all drawn from prior literature; no ad hoc assumptions or fitted parameters are introduced. The central new ingredient is the pointwise algebraic inequality (Lemma 2.1), which is proven within the paper.

assumptions (5)

domain assumption Known C^{1,α} regularity and convergence of the elliptic regularization (1.9): Du_ε ∈ L∞ uniformly and Du_ε → Du in C^{0,α}.
Invoked in Section 3 immediately after (1.9), citing [33, 26, 10]. The proof of Theorem 1.1 requires identifying the weak limit of the regularized gradients, which relies on this convergence.
domain assumption Parabolic regularization theory for (1.12) and (1.13): existence of smooth viscosity/weak solutions u_ε with u_ε → u in C^0 and Du_ε ∈ L∞ uniformly.
Invoked in Section 4 for (1.12), citing [22], and in Section 5 for (1.13), citing [11, 36]. Needed to pass from uniform estimates on u_ε to estimates on the limiting solution u.
standard math Gehring's lemma (self-improvement of reverse Hölder inequalities).
Used in the proofs of Corollary 1.2 and Theorem 1.3 to upgrade L^2 estimates to L^{2+δ} integrability.
standard math Parabolic Sobolev-Poincaré inequality and compact embedding theorems.
Used in Section 4 to obtain the reverse Hölder inequality leading to Gehring's lemma, and in Sections 3 and 5 for compactness and limit identification.
standard math Equivalence of weak and viscosity solutions for the p-Laplace equations.
Used to state the theorems uniformly for 'weak/viscosity' solutions, citing [20, 21, 31].

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Pith. "Pith review of Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality." pith.science (2026). https://pith.science/paper/J34AEHOI

@misc{pith2026190801547,
  author       = {Pith},
  title        = {Pith review of: Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J34AEHOI}},
  note         = {Machine review of arXiv:1908.01547}
}

abstract

Denote by $\Delta$ the Laplacian and by $\Delta_\infty $ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $\Delta v\Delta_\infty v$: for every $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {\Delta v \Delta_\infty v } -\frac12[|D^2v|^2-(\Delta v)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results: 1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have $$|Du|^{\frac{p-\gamma}2}Du\in W^{1,2}_{\rm loc},$$ with $\gamma<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the known $W^{2,q}_{\rm loc}$-regularity of $p$-harmonic functions for some $q>2$. 2. When $n\ge 2$ and $p\in(1,2)\cup(2,3+\frac2{n-2})$, the viscosity solutions to parabolic normalized $p $-Laplace equation have the $W_{\rm loc}^{2,q}$-regularity in the spatial variable and the $W_{\rm loc}^{1,q}$-regularity in the time variable for some $q>2$. Especially, when $n=2$ an open question in [17] is completely answered. 3. When $n\ge 1 $ and $p\in(1,2)\cup(2,3)$, the weak/viscosity solutions to parabolic $p $-Laplace equation have the $W_{\rm loc}^{2,2}$-regularity in the spatial variable and the $W_{\rm loc}^{1,2}$-regularity in the time variable. The range of $p$ (including $p=2$ from the classical result) here is sharp for the $W_{\rm loc}^{2,2}$-regularity.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [22]

Jin and L

T. Jin and L. Silvestre, H¨ older gradient estimates for parabolic homogeneous p-La placian equations. J. Math. Pures Appl. 108 (2017), 63-87

work page 2017

[2] [1]

Attouchi and M

A. Attouchi and M. Parviainen, H¨ older regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian. Commun. Contemp. Math. 20 (2018), 1750035

work page 2018

[3] [2]

Attouchi and E

A. Attouchi and E. Ruosteenoja, Remarks on regularity for p-laplacian type equations in non-divergence form. J. Diﬀ. Equ. 265 (2018), 1922-1961

work page 2018

[4] [3]

Bers and L

L. Bers and L. Nirenberg, On linear and non-linear elliptic boundary value problems i n the plane. Convegno Internazionale sulle Equazioni Lineari alle Deri vate Parziali, Trieste, 1954, pp. 141-167, Edizioni Cremonese, Roma, 1955

work page 1954

[5] [4]

Bojarski and T

B. Bojarski and T. Iwaniec, p-harmonic equation and quasiregular mappings. Partial diﬀer- ential equations (Warsaw, 1984), Banach Center Publ, 19, PW N, Warsaw (1987) 25-38

work page 1987

[6] [5]

Banerjee and N

A. Banerjee and N. Garofalo, Gradient bounds and monotonicity of the energy for some nonlinear singular diﬀusion equations . Indiana Univ. Math. J. 62 (2013), 699-736

work page 2013

[7] [6]

Banerjee and N

A. Banerjee and N. Garofalo, On the Dirichlet boundary value problem for the normalized p-Laplacian evolution. Commun. Pure Appl. Anal., 14 (2015), 1-21

work page 2015

[8] [7]

L. A. Caﬀarelli and X. Cabr´ e, Fully nonlinear elliptic eq uations. volume 43 of American Mathematical Society Colloquium Publications. American M athematical Society, Provi- dence, RI, 1995

work page 1995