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Pogorelov type interior $C^2$ estimate for Hessian quotient equation and its application

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Pith's one-line read The paper proves a Pogorelov-type interior $C^2$ estimate for the Hessian quotient equation $\sigma_n/\sigma_k(D^2u)=f$ and uses it to upgrade convex viscosity solutions with sharp Hölder or Sobolev regularity to $C^{3,\beta}$.

desk verdict A promising attack on a real regularity problem, but the submitted proof has a load-bearing gap at Lemma 5.1 that the authors do not fix; it deserves peer review, not desk rejection, and not citation yet. read the letter →

arxiv 2505.10287 v1 pith:HTU3NEJN submitted 2025-05-15 math.AP

Siyuan Lu , Yi-Lin Tsai This is my paper · ORCID
classification math.AP MSC 35J6035J1535J96
keywords HessianquotientequationPogorelov-typeestimateinteriorC^2Legendretransformviscositysolutionstrictconvexitysharpregularityexponentselementarysymmetricfunctions
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 a Pogorelov-type interior $C^2$ estimate for the Hessian quotient equation $\frac{\sigma_n}{\sigma_k}(D^2u)=f$: for a convex solution that vanishes with zero gradient at the origin and equals $2$ on the boundary of its section $\Sigma_2$, the second derivatives are bounded in a fixed ball $B_\tau\subset\Sigma_1$, with constants depending only on $n,k$, the diameter, the $C^{0,1}$ norm, and the data of $f$. Such an estimate was missing for quotient equations, and it is what upgrades weak convex solutions to classical ones. As an application, the paper proves that convex viscosity solutions with $u\in C^{1,\alpha}$ for $\alpha>1-\frac{2}{n-k}$, or with $u\in W^{2,p}$ for $p\ge\frac{(n-1)(n-k)}2$, are $C^{3,\beta}$ in the interior whenever $k\le n-3$. Both thresholds are sharp, matching the singular example cited as [23] in the paper. A broader consequence is a new route to lower bounds on Hessian eigenvalues, which are usually harder to obtain than upper bounds.

What carries the argument

The load-bearing mechanism is the Legendre transform $w(y)=\sup_x(x\cdot y-u(x))$, which converts the quotient equation into the Hessian equation $\sigma_k(D^2w)=f^{-1}$ on the dual section. The log of the largest Hessian eigenvalue, $b=\ln\lambda_1$, is shown to satisfy a Jacobi inequality $\sum_i F_{ii}b_{ii}\ge c\sum_i F_{ii}b_i^2-C$ using a concavity inequality for Hessian quotient operators; after the transform this becomes a subsolution inequality $\sum_i G_{ii}b^*_{ii}\ge -C$ for a uniformly elliptic operator whose coefficients are $G_{ii}=F^{-2}F_{ii}u_{ii}^2$. The new strict-positivity estimate of Lemma 4.1, $(-u)^\beta/\lambda_{\min}\le C$, obtained by a Pogorelov-type argument on the transformed equation, is what supplies the uniform ellipticity and the lower eigenvalue bound used in the mean value inequality and in the final integration by parts with $H_{ij}=\sigma_kF_{ij}$.

What would settle it

Take the singular convex solution from the paper's reference [23], where $n-k\ge3$, and compute the transformed coefficients $G_{ii}=F^{-2}F_{ii}u_{ii}^2$ on a sequence of points where the largest eigenvalue of $D^2u$ diverges; if $G_{ii}$ fails to remain bounded, the uniform ellipticity asserted in (5.1) is false and the mean value inequality in Lemma 5.2 has no basis.

Watch

Extended reading notes

Core claim

On its own terms, the central claim is Theorem 1.2: for $n\ge2$, $1\le k<n$, a positive $f\in C^{1,1}(\Sigma_2\times\mathbb{R})$, and a convex $C^4$ solution $u$ of $\frac{\sigma_n}{\sigma_k}(D^2u)=f(x,u)$ with $u(0)=0$, $Du(0)=0$, and $u=2$ on $\partial\Sigma_2$, there are constants $\tau>0$ and $C$, depending only on the stated data, such that $B_\tau\subseteq\Sigma_1$ and $\max_{B_\tau}|D^2u|\le C$. From this a priori bound the paper derives Theorem 1.4: under either of the two sharp regularity assumptions, any convex viscosity solution is in $C^{3,\beta}(\Omega)$ for every $\beta<1$, with quantitative estimates on compact subdomains. The proof is designed so that the same estimate also forces the Hessian of $u$ to be uniformly positive in an interior ball, a strict-convexity property that the regularity argument needs.

Load-bearing premise

The load-bearing premise is that the lower-bound estimate on the Hessian of $u$ in Lemma 4.1 also gives a uniform upper bound on the transformed coefficients $G_{ii}$; that upper bound is equivalent to the uniform upper bound on $D^2u$ that Theorem 1.2 is trying to prove, and the paper does not supply an independent argument for it.

Editorial extensions

If this is right

  • For $k\le n-3$, convex viscosity solutions with $u\in C^{1,\alpha}$ and $\alpha>1-\frac{2}{n-k}$ are $C^{3,\beta}$ with interior estimates depending on the natural data.
  • For the same range, convex viscosity solutions with $u\in W^{2,p}$ and $p\ge\frac{(n-1)(n-k)}2$ are also $C^{3,\beta}$, including the critical borderline case $p=\frac{(n-1)(n-k)}2$.
  • The exponents cannot be improved: the singular example cited as [23] shows that the borderline cases are genuinely singular, so the hypotheses of Theorem 1.4 are optimal.
  • The Pogorelov-type estimate itself holds for all $n\ge2$ and all $1\le k<n$ under section boundary conditions, extending earlier results that only covered $n-k\le2$.
  • The proof gives a new way to obtain uniform lower bounds on the smallest Hessian eigenvalue, a quantity that is normally much harder to control than upper bounds.

Reading between the lines

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

  • The same Legendre-transform and strict-positivity strategy may apply to general Hessian quotient equations $\sigma_k/\sigma_l=f$ with $k-l\le2$, where pure interior $C^2$ estimates remain open; the equation treated here is the special case with $l=k$ and $k=n$.
  • A reader wanting to test the method should check whether the strict positivity estimate can be pushed to the borderline $k=n-2$; if it can, the regularity theorem would extend to that case, which is currently excluded.
  • The critical Sobolev exponent $p=\frac{(n-1)(n-k)}2$ suggests an interpolation phenomenon: at exactly this exponent the slicing argument used in the borderline case produces the needed lower bound on the section, while below it the singular example takes over; identifying the analogue for other quotient equations would clarify the boundary between regular and singular regimes.
  • If the regularity theorem is combined with standard localization arguments, it may extend to equations with $f$ depending on $x$ and $u$ or to nonconstant boundary data, since the a priori estimate has such data in its constants.
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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

2 major / 3 minor

Summary. The paper claims a Pogorelov-type interior C^2 estimate for convex solutions of the Hessian quotient equation sigma_n/sigma_k(D^2u)=f on sections with homogeneous boundary data (Theorem 1.2), and derives C^{3,beta} regularity for convex viscosity solutions under sharp C^{1,alpha} or W^{2,p} assumptions (Theorem 1.4). The strategy is to use the Legendre transform: Lemma 4.1 gives a lower bound on the smallest eigenvalue of D^2u; Lemma 3.3 (Jacobi inequality) yields an inequality for b = ln lambda_1; Lemma 5.1 claims that in Legendre coordinates b is a subsolution of a uniformly elliptic operator; Lemma 5.2 applies the local maximum principle; and Section 6 bounds the integral of b by integration by parts. The regularity applications follow the work of Urbas and of Collins and Mooney.

Significance. If Theorem 1.2 were established, the paper would be a significant contribution: Pogorelov-type estimates for general Hessian quotient equations have been missing, and the sharp regularity thresholds in Theorem 1.4 would match the example in [23]. The Legendre-transform approach to lower Hessian bounds (Lemma 4.1) and the induction lemma for integration by parts (Lemma 6.1) are interesting and potentially useful. The paper is clearly organized and carefully states the dependencies of constants. However, the central Lemma 5.1 contains an unjustified uniform ellipticity assertion, and the proof relies on a concavity inequality that is only cited to a private communication. As written, the main estimate is not supported.

major comments (2)
  1. [Lemma 5.1, display (5.1)] The two-sided bound 1/C <= G_ii <= C is not a consequence of Lemma 4.1. Lemma 4.1 gives (-u*)^beta * mu_1 <= C, and in (Sigma_1)* we have -u* >= 1, so mu_1 <= C. Since D^2w is positive semidefinite, this yields only the upper bound G_ii <= C. The lower bound G_ii >= 1/C is equivalent to a lower bound on every eigenvalue of D^2w, hence to an upper bound on D^2u, which is precisely the conclusion of Theorem 1.2 that the proof is meant to establish. The chain of inequalities immediately before (5.1) needs -C * sum_i 1/G_ii - C >= -C', which is exactly the missing lower bound. Consequently the uniform ellipticity used in Lemma 5.2 (via the local maximum principle) and in the integration by parts in Section 6 rests on an unproved and effectively circular premise. No argument bridging mu_1 <= C to full control of the coefficients of sum_i G_ii d_ii is supplied.
  2. [Lemma 3.2] The concavity inequality for the Hessian quotient operator is stated without proof and is referenced only to [14], which is listed in the bibliography as a private communication. Lemma 3.3 and therefore the Jacobi inequality on which the entire proof of Theorem 1.2 depends, rely on this unpublished result. The manuscript should either include a complete proof of Lemma 3.2 or cite a publicly available source that contains its proof (reference [34], if it contains the statement, is not cited in Lemma 3.2). As written, a central ingredient of the main theorem is not verifiable from the manuscript or from a citable reference.
minor comments (3)
  1. [Lemma 4.1, statement] The displayed conclusion appears as '(-u)^beta * lambda_min <= C', but the proof shows that the quantity to be bounded is (-u*)^beta * mu_1, which equals (-u)^beta / lambda_min. The denominator is missing in the displayed statement.
  2. [Lemma 5.2, proof] The sentence 'By (5.1), sum_i G_ii >= C' uses the same letter C for the constant in (5.2) and for the lower bound; the argument requires a quantitatively chosen constant, for instance writing sum_i G_ii (b* + Lambda |y|^2)_ii >= 0 with Lambda chosen after the constants in (5.1)-(5.2). As written this is a notational shortcut rather than a mathematical error.
  3. [Lemma 3.3, proof] The proof is omitted with the statement 'the rest are exactly the same' as the proof of Lemma 4.1 in [23]. Since [23] is a preprint by the first author, the manuscript would be more self-contained if the argument were included or if the precise dependence on results of [23] were spelled out.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central a priori estimate is derived from the transformed equation; the only self-citation is a routine technical lemma, not a reduction to the theorem's conclusion.

full rationale

The paper's derivation chain is: Jacobi inequality (Lemma 3.3) -> Legendre-transformed subsolution property (Lemma 5.1) -> mean value inequality (Lemma 5.2) -> integration by parts (Section 6) -> Theorem 1.2. No fitted parameters appear, and the final bound on |D^2u| is obtained by bounding an integral of b = ln(lambda_1)+K_0, not by inserting the desired bound as an input. The one self-citation is Lemma 3.3, whose proof is omitted with the note "The proof is almost the same as the proof of Lemma 4.1 in [23]"; [23] is an earlier independent paper by the first author, and the present paper supplies the modified concavity ingredient (Lemma 3.2). The apparently circular-looking line in Lemma 5.1, "By Lemma 4.1, |D^2w| is bounded in (Sigma_1)^*. It follows that 1/C <= G_ii <= C", is terse but not a reduction to the conclusion: Lemma 4.1 gives an upper bound on the largest eigenvalue of D^2w, and the transformed equation sigma_{n-k}(D^2w)=f^* with f^* bounded below, together with the elementary identity sigma_m(mu|i) <= C mu_1 sigma_{m-1}(mu|i), supplies the missing lower bound on sigma_{n-k-1}(mu|i). No uniqueness theorem is imported, no known result is merely renamed, and the sharpness example is cited from [23] rather than being derived from the present theorem. The proof is self-contained modulo a routine technical self-citation, so the circularity score is low.

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

No empirical or fitted parameters appear in this analytic proof. The listed axioms include the unpublished concavity inequality and the unproved lower-bound assertion on G_ii, which are the main external or ad hoc inputs on which the central estimate rests.

assumptions (4)
  • domain assumption The Guan-Sroka concavity inequality (Lemma 3.2) is valid and applies to F=sigma_n/sigma_k.
    The Jacobi inequality in Lemma 3.3 is built on this inequality; reference [14] is a private communication with no public proof in the paper.
  • ad hoc to paper The transformed coefficients G_ii are uniformly bounded below in (Sigma_1)^*.
    Asserted in (5.1) and used for the local maximum principle; Lemma 4.1 does not establish it, and no other justification is provided.
  • standard math Standard Newton-Maclaurin and Garding cone identities hold for elementary symmetric functions.
    Used throughout, for example in Lemmas 2.1, 6.1, and the proof of Theorem 1.2.
  • standard math The viscosity-to-distribution conversion for the Jacobi inequality holds via Theorem 1 in [16].
    Used in Section 6 to integrate the Jacobi inequality by parts.

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Pith. "Pith review of Pogorelov type interior $C^2$ estimate for Hessian quotient equation and its application." pith.science (2026). https://pith.science/paper/HTU3NEJN

@misc{pith2026250510287,
  author       = {Pith},
  title        = {Pith review of: Pogorelov type interior $C^2$ estimate for Hessian quotient equation and its application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTU3NEJN}},
  note         = {Machine review of arXiv:2505.10287}
}
abstract

In this paper, we derive a Pogorelov type interior $C^2$ estimate for the Hessian quotient equation $\frac{\sigma _n}{\sigma _k}\left( D^2u\right) =f$. As an application, we show that convex viscosity solutions are regular for $k\leq n-3$ if $u\in C^{1,\alpha}$ with $\alpha>1-\frac{2}{n-k}$ or $u\in W^{2,p}$ with $p\geq\frac{(n-1)(n-k)}{2}$. Both exponents are sharp in view of the example in arXiv:2401.12229.

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Works this paper leans on

43 extracted references · 38 canonical work pages

  1. [23]

    Lu, Interior C 2 estimate for Hessian quotient equation in general dimension , arXiv:2401.12229

    S. Lu, Interior C 2 estimate for Hessian quotient equation in general dimension , arXiv:2401.12229

    arXiv

  2. [34]

    Remarks on Hessian quotient equations on Riemannian manifolds

    M. Sroka, Remarks on Hessian quotient equations on Riemannian manifolds , arXiv:2501.03386

    work page Pith review arXiv

  3. [14]

    Guan and M

    P. Guan and M. Sroka, A special concavity property for positive Hessian quotient operators , private communication

    work page

  4. [1]

    Bao and J

    J. Bao and J. Chen, Optimal regularity for convex strong solutions of special Lagrangian equa- tions in dimension 3, Indiana Univ. Math. J. 52 (2003), no. 5, 1231–1249

    work page 2003

  5. [2]

    J. Bao, J. Chen, B. Guan and M. Ji, Liouville property and regularity of a Hessian quotient equation, Amer. J. Math. 125 (2003), no. 2, 301–316

    work page 2003

  6. [3]

    Brendle, K

    S. Brendle, K. Choi and P. Daskalopoulos, Asymptotic behavior of flows by powers of the Gauss- ian curvature, Acta Math. 219 (2017), no. 1, 1–16

    work page 2017

  7. [4]

    L. A. Caffarelli and X. Cabr´ e,Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 1995. vi+104 pp

    work page 1995

  8. [5]

    L. A. Caffarelli and C. E. Guti´ errez,Properties of the solutions of the linearized Monge-Amp` ere equation, Amer. J. Math. 119 (1997), no. 2, 423–465

    work page 1997

Show all 43 references
  1. [6]

    C. Chen, F. Han and Q. Ou, The interior C 2 estimate for the Monge-Amp` ere equation in dimension n = 2, Anal. PDE 9 (2016), no. 6, 1419–1432

    2016

  2. [7]

    J. Chen, M. Warren and Y. Yuan, A priori estimate for convex solutions to special Lagrangian equations and its application , Comm. Pure Appl. Math. 62 (2009), no. 4, 583–595

    2009

  3. [8]

    R. Chen, H. Jian and Z. Zhou, An Integral Approach to Prescribing Scalar Curvature Equations, arXiv:2408.14850

    arXiv

  4. [9]

    Chou and X.-J

    K.-S. Chou and X.-J. Wang, A variational theory of the Hessian equation , Comm. Pure Appl. Math. 54 (2001), no. 9, 1029–1064

    2001

  5. [10]

    T. C. Collins and C. Mooney, Dimension of the minimum set for the real and complex Monge- Amp` ere equations in critical Sobolev spaces, Anal. PDE 10 (2017), no. 8, 2031–2041

    2017

  6. [11]

    M. G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67

    1992

  7. [12]

    Guan and G

    P. Guan and G. Qiu, Interior C 2 regularity of convex solutions to prescribing scalar curvature equations, Duke Math. J. 168 (2019), no. 9, 1641–1663

    2019

  8. [13]

    P. Guan, C. Ren and Z. Wang, GlobalC 2-estimates for convex solutions of curvature equations , Comm. Pure Appl. Math. 68 (2015), no. 8, 1287–1325

    2015

  9. [15]

    Heinz, On elliptic Monge-Amp` ere equations and Weyl’s embedding problem, J

    E. Heinz, On elliptic Monge-Amp` ere equations and Weyl’s embedding problem, J. Analyse Math. 7 (1959), 1–52

    1959

  10. [16]

    Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial

    H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac. 38 (1995), no. 1, 101–120. 34 SIYUAN LU AND YI-LIN TSAI

    1995

  11. [17]

    Ishii and P.-L

    H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial dif- ferential equations, J. Differential Equations 83 (1990), no. 1, 26–78

    1990

  12. [18]

    M. Li, C. Ren and Z. Wang, An interior estimate for convex solutions and a rigidity theorem , J. Funct. Anal. 270 (2016), no. 7, 2691–2714

    2016

  13. [19]

    Lin and N

    M. Lin and N. S. Trudinger, The Dirichlet problem for the prescribed curvature quotient equa- tions, Topol. Methods Nonlinear Anal. 3 (1994), no. 2, 307–323

    1994

  14. [20]

    Liu, Interior C 2 estimate for Monge-Amp` ere equation in dimension two, Proc

    J. Liu, Interior C 2 estimate for Monge-Amp` ere equation in dimension two, Proc. Amer. Math. Soc. 149 (2021), no. 6, 2479–2486

    2021

  15. [21]

    Liu and J

    S. Liu and J. Bao, The local regularity for strong solutions of the Hessian quotient equation , J. Math. Anal. Appl. 303 (2005), no. 2, 462–476

    2005

  16. [22]

    Lu, InteriorC 2 estimate for Hessian quotient equation in dimension three , arXiv:2311.05835

    S. Lu, InteriorC 2 estimate for Hessian quotient equation in dimension three , arXiv:2311.05835

    arXiv

  17. [24]

    McGonagle, C

    M. McGonagle, C. Song and Y. Yuan, Hessian estimates for convex solutions to quadratic Hessian equation, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 36 (2019), no. 2, 451–454

    2019

  18. [25]

    Mooney, Partial Regularity for Singular Solutions to the Monge-Amp` ere Equation , Comm

    C. Mooney, Partial Regularity for Singular Solutions to the Monge-Amp` ere Equation , Comm. Pure Appl. Math. 68 (2015), no. 6, 1066–1084

    2015

  19. [26]

    Mooney, Strict 2-convexity of convex solutions to the quadratic Hessian equation, Proc

    C. Mooney, Strict 2-convexity of convex solutions to the quadratic Hessian equation, Proc. Amer. Math. Soc. 149 (2021), no. 6, 2473–2477

    2021

  20. [27]

    A. V. Pogorelov, The Minkowski multidimensional problem , Translated from the Russian by Vladimir Oliker. Introduction by Louis Nirenberg. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, DC; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. 106 pp

    1978

  21. [28]

    Qiu, Interior Hessian estimates for Sigma- 2 equations in dimension three , Front

    G. Qiu, Interior Hessian estimates for Sigma- 2 equations in dimension three , Front. Math. 19 (2024), no. 4, 577–598

    2024

  22. [29]

    Qiu, Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimen- sion three, Amer

    G. Qiu, Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimen- sion three, Amer. J. Math. 146 (2024), no. 3, 579–605

    2024

  23. [30]

    Shankar and Y

    R. Shankar and Y. Yuan, Hessian estimate for semiconvex solutions to the sigma- 2 equation, Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 30, 12 pp

    2020

  24. [31]

    Shankar and Y

    R. Shankar and Y. Yuan, Regularity for the Monge-Amp` ere equation by doubling, Math. Z. 307 (2024), no. 2, Paper No. 34, 7 pp

    2024

  25. [32]

    Shankar and Y

    R. Shankar and Y. Yuan, Hessian estimates for the sigma- 2 equation in dimension four , Ann. of Math. (2) 201 (2025), no. 2, 489–513

    2025

  26. [33]

    Sheng, J

    W. Sheng, J. I. E. Urbas and X.-J. Wang, Interior curvature bounds for a class of curvature equations, Duke Math. J. 123 (2004), no. 2, 235–264

    2004

  27. [35]

    N. S. Trudinger, On the Dirichlet problem for Hessian equations , Acta Math. 175 (1995), no. 2, 151–164

    1995

  28. [36]

    Tu, Pogorelov type estimates for (n− 1)-Hessian equations and related rigidity theorems , arXiv:2405.02939

    Q. Tu, Pogorelov type estimates for (n− 1)-Hessian equations and related rigidity theorems , arXiv:2405.02939

    arXiv

  29. [37]

    J. I. E. Urbas, Regularity of generalized solutions of Monge-Amp` ere equations, Math. Z. 197 (1988), no. 3, 365–393

    1988

  30. [38]

    J. I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J. 39 (1990), no. 2, 355–382

    1990

  31. [39]

    Wang and Y

    D. Wang and Y. Yuan, Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions , Amer. J. Math. 136 (2014), no. 2, 481–499

    2014

  32. [40]

    Warren and Y

    M. Warren and Y. Yuan, Hessian estimates for the sigma- 2 equation in dimension 3, Comm. Pure Appl. Math. 62 (2009), no. 3, 305–321

    2009

  33. [41]

    Yuan, A monotonicity approach to Pogorelov’s Hessian estimates for Monge-Amp` ere equa- tion, Math

    Y. Yuan, A monotonicity approach to Pogorelov’s Hessian estimates for Monge-Amp` ere equa- tion, Math. Eng. 5 (2023), no. 2, Paper No. 037, 6 pp

    2023

  34. [42]

    Zhang, C 2 estimates for k-Hessian equations and a rigidity theorem , arXiv:2408.10781

    R. Zhang, C 2 estimates for k-Hessian equations and a rigidity theorem , arXiv:2408.10781. HESSIAN QUOTIENT EQUATION 35

    arXiv

  35. [43]

    Zhou, Notes on generalized special Lagrangian equation, Calc

    X. Zhou, Notes on generalized special Lagrangian equation, Calc. Var. Partial Differential Equa- tions 63 (2024), no. 8, Paper No. 197, 28 pp. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada. Email address : s...

    2024

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