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arxiv: 2606.07203 · v1 · pith:NTMMSWROnew · submitted 2026-06-05 · 🧮 math.AP

Gradient Regularity for Fully Nonlinear Equations with Variable Degeneracy and Hamiltonian Lower-Order Terms

Gleiciano Cosmo , Rafael R. Costa , Diego Marcon This is my paper

Pith reviewed 2026-06-27 21:39 UTC · model grok-4.3

classification 🧮 math.AP MSC 35J6035B65
keywords viscosity solutionsfully nonlinear elliptic equationsgradient regularityvariable degeneracyHölder estimatesimprovement of flatnessSchauder estimates
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The pith

Viscosity solutions to fully nonlinear equations with variable gradient degeneracy have interior Hölder continuous gradients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that viscosity solutions to the equation |∇u|^{p(x)} F(∇²u) + a(x)|∇u|^{q(x)} = f(x) are C^{1,α} in the interior, where the exponent α is controlled by the largest value of the degeneracy exponent p and by the regularity known for the associated homogeneous equation F(∇²v) = 0. This regularity extends to pointwise improvements at locations where f and a vanish at controlled rates, and to a separation estimate at extremal points showing the solution deviates from its maximum or minimum by an order strictly higher than two. The results matter because they supply gradient control for a class of equations whose ellipticity strength changes with both position and the size of the gradient itself.

Core claim

The authors establish interior Hölder estimates for the gradient of viscosity solutions, with the Hölder exponent fixed by the maximal degeneracy rate of p(x) and by the regularity available for the homogeneous uniformly elliptic problem. They obtain further pointwise Hölder improvements at points where the source f and the coefficient a vanish with given Hölder moduli. At extremal points they prove a Schauder-type estimate in which the solution separates from its extremal value with order strictly larger than two. The proofs rely on compactness for shifted equations, stability of viscosity solutions, and improvement-of-flatness iterations.

What carries the argument

improvement-of-flatness iterations applied to shifted equations, using compactness estimates and viscosity stability

If this is right

  • The gradient is Hölder continuous in the interior with an explicit exponent depending on max p and the homogeneous problem.
  • At points where f and a vanish with prescribed Hölder rates the solution gains additional pointwise regularity.
  • Near extremal points the solution separates from its extremal value with order strictly greater than two.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same compactness-plus-iteration strategy may extend to equations whose degeneracy depends on |∇u| through more general functions than power laws.
  • The separation estimate at extrema could be used to obtain strict convexity or concavity conclusions for solutions of related geometric problems.
  • Numerical approximation schemes for these equations could exploit the Hölder modulus to prove convergence rates without additional structural assumptions.

Load-bearing premise

The operator F stays uniformly elliptic even when the gradient vanishes or becomes large.

What would settle it

A concrete viscosity solution whose gradient fails to be Hölder continuous with the predicted exponent at an interior point where p attains its maximum, despite F remaining uniformly elliptic.

read the original abstract

We study local regularity properties of viscosity solutions to fully nonlinear elliptic equations with variable gradient degeneracy and Hamiltonian-type lower-order terms, \[ |\nabla u|^{p(x)}F(\nabla^{2}u) + a(x)|\nabla u|^{q(x)} = f(x). \] Here, $F$ is uniformly elliptic, while the exponents $p$ and $q$ are allowed to vary in space. We prove interior H\"older estimates for the gradient, with an exponent determined by the maximal degeneracy rate and by the regularity available for the associated homogeneous uniformly elliptic equation. We also obtain pointwise improvements at points where the source term and the Hamiltonian coefficient vanish with prescribed H\"older rates. Finally, at extremal points, we establish a Schauder-type estimate showing that the solution separates from its extremal value with order strictly larger than two. The proofs combine compactness estimates for shifted equations, stability of viscosity solutions, and improvement-of-flatness iterations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript establishes interior Hölder gradient estimates for viscosity solutions of the fully nonlinear equation |∇u|^{p(x)} F(∇²u) + a(x)|∇u|^{q(x)} = f(x), where F is uniformly elliptic and the exponents p(x), q(x) are allowed to vary spatially. The Hölder exponent is determined by the maximal degeneracy rate together with the regularity of the associated homogeneous uniformly elliptic equation. Additional claims include pointwise improvements of the solution at points where f and a vanish at prescribed Hölder rates, and a Schauder-type estimate at extremal points showing that the solution separates from its extremal value with order strictly larger than two. The proofs rely on compactness estimates for shifted equations, stability of viscosity solutions, and improvement-of-flatness iterations.

Significance. If the results hold, the work extends gradient regularity theory for degenerate fully nonlinear equations to the setting of spatially varying degeneracy rates and Hamiltonian lower-order terms. The approach of reducing to the worst-case degeneracy while retaining uniform ellipticity of F is consistent with existing viscosity-solution techniques and could apply to models with heterogeneous degeneracy. The combination of compactness, stability, and improvement-of-flatness is a standard and reproducible strategy when the uniform ellipticity assumption is satisfied.

minor comments (1)
  1. The abstract refers to 'the regularity available for the associated homogeneous uniformly elliptic equation' without specifying the precise Hölder or C^{1,α} assumptions on the coefficients of that equation; a brief clarification in the introduction would help readers identify the exact dependence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation to accept the manuscript. The report accurately summarizes the main contributions regarding interior Hölder gradient estimates, pointwise improvements, and Schauder-type estimates for the degenerate fully nonlinear equation with variable exponents.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's claimed results rest on standard compactness estimates for shifted equations, stability of viscosity solutions, and improvement-of-flatness iterations, all drawn from established external viscosity theory under the assumption of uniform ellipticity of F. The Hölder exponent is explicitly determined by the maximal degeneracy rate together with regularity of the associated homogeneous equation, without any reduction to fitted parameters, self-definitions, or self-citation chains within the paper. No load-bearing step reduces by construction to the inputs, and the variable exponents p(x), q(x) are incorporated via worst-case analysis consistent with the stated methods.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard background assumptions from viscosity solution theory for uniformly elliptic operators; no free parameters, invented entities, or ad-hoc axioms are introduced in the provided text.

axioms (1)
  • domain assumption F is uniformly elliptic
    Invoked in the abstract as the structural hypothesis enabling compactness and stability arguments.

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Works this paper leans on

38 extracted references · 20 canonical work pages

  1. [1]

    M. D. Amaral, J. V. da Silva, G. C. Ricarte, and R. Teymurazyan,Sharp regularity estimates for quasilinear evolution equations, Israel J. Math.231(2019), no. 1, 25–45. DOI: 10.1007/s11856-019-1842-1

    work page doi:10.1007/s11856-019-1842-1 2019

  2. [2]

    P. D. S. Andrade, D. S. dos Prazeres, and M. S. Santos,Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy, Nonlinearity37(2024), no. 4, Paper No. 045009. DOI: 10.1088/1361- 6544/ad2c22

    work page doi:10.1088/1361- 2024

  3. [3]

    P. D. S. Andrade and T. M. Nascimento,Optimal regularity for degenerate elliptic equations with Hamiltonian terms, arXiv:2508.03924, 2025

    arXiv 2025

  4. [4]

    D. J. Ara´ ujo, R. Leit˜ ao, and E. V. Teixeira,Infinity Laplacian equation with strong absorptions, J. Funct. Anal. 270(2016), no. 6, 2249–2267. DOI: 10.1016/j.jfa.2015.12.012

    work page doi:10.1016/j.jfa.2015.12.012 2016

  5. [5]

    D. J. Ara´ ujo, G. Ricarte, and E. V. Teixeira,Geometric gradient estimates for solutions to degenerate elliptic equations, Calc. Var. Partial Differential Equations53(2015), no. 3-4, 605–625

    2015

  6. [6]

    D. J. Ara´ ujo, G. C. Ricarte, and E. V. Teixeira,Singularly perturbed equations of degenerate type, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire34(2017), no. 3, 655–678. DOI: 10.1016/j.anihpc.2016.03.004

    work page doi:10.1016/j.anihpc.2016.03.004 2017

  7. [7]

    D. J. Ara´ ujo, E. V. Teixeira, and J. M. Urbano,A proof of theC p′ -regularity conjecture in the plane, Adv. Math. 316(2017), 541–553. DOI: 10.1016/j.aim.2017.06.027

    work page doi:10.1016/j.aim.2017.06.027 2017

  8. [8]

    D. J. Ara´ ujo, E. V. Teixeira, and J. M. Urbano,Towards theC p′ -regularity conjecture in higher dimensions, Int. Math. Res. Not. IMRN2018, no. 20, 6481–6495. DOI: 10.1093/imrn/rnx068

    work page doi:10.1093/imrn/rnx068

  9. [9]

    D. J. Ara´ ujo, R. Teymurazyan, and J. M. Urbano,Geometric properties of free boundaries in degenerate quenching problems, SIAM J. Math. Anal., to appear. arXiv:2402.11536

    arXiv

  10. [10]

    D. J. Ara´ ujo, R. Teymurazyan, and V. Voskanyan,Sharp regularity for singular obstacle problems, Math. Ann. 387(2023), no. 3–4, 1367–1401

    2023

  11. [11]

    E. C. Bezerra J´ unior, J. V. da Silva, G. C. Rampasso, and G. C. Ricarte,Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy, J. Lond. Math. Soc. (2)108(2023), no. 2, 622–665

    2023

  12. [12]

    Birindelli and F

    I. Birindelli and F. Demengel,Comparison principle and Liouville type results for singular fully nonlinear opera- tors, Ann. Fac. Sci. Toulouse Math. (6)13(2004), no. 2, 261–287

    2004

  13. [13]

    Birindelli and F

    I. Birindelli and F. Demengel,Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Commun. Pure Appl. Anal.6(2007), no. 2, 335–366

    2007

  14. [14]

    Birindelli and F

    I. Birindelli and F. Demengel,Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains, J. Math. Anal. Appl.352(2009), no. 2, 822–835

    2009

  15. [15]

    Birindelli and F

    I. Birindelli and F. Demengel,Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations249(2010), no. 5, 1089–1110

    2010

  16. [16]

    Birindelli and F

    I. Birindelli and F. Demengel,Regularity for radial solutions of degenerate fully nonlinear equationsNonlinear Anal.75(2012), no. 17, 6237–6249

    2012

  17. [17]

    Birindelli and F

    I. Birindelli and F. Demengel,C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM: Control, Optimisation and Calculus of Variations,20(2014), no. 4, 1009–1024. DOI: 10.1051/cocv/2014005. GRADIENT REGULARITY WITH VARIABLE DEGENERACY 25

    work page doi:10.1051/cocv/2014005 2014

  18. [18]

    Birindelli and F

    I. Birindelli and F. Demengel,Fully nonlinear operators with Hamiltonian: H¨ older regularity of the gradient, NoDEA Nonlinear Differential Equations Appl.23(2016), no. 4, Art. 41, 17 pp

    2016

  19. [19]

    A. C. Bronzi, E. A. Pimentel, G. C. Rampasso, and E. V. Teixeira,Regularity of solutions to a class of variable- exponent fully nonlinear elliptic equations, J. Funct. Anal.279(2020), no. 12, Paper No. 108781, 31 pp. DOI: 10.1016/j.jfa.2020.108781

    work page doi:10.1016/j.jfa.2020.108781 2020

  20. [20]

    Caffarelli and X

    L. Caffarelli and X. Cabr´ e, Fully Nonlinear Elliptic Equations.American Mathematical Society Colloquium Pub- lications, 43, American Mathematical Society, Providence, RI, 1995

    1995

  21. [21]

    Challal, A

    S. Challal, A. Lyaghfouri, J. F. Rodrigues, and R. Teymurazyan,On the regularity of the free boundary for quasilinear obstacle problems, Interfaces Free Bound.16(2014), no. 3, 359–394. DOI: 10.4171/IFB/323

    work page doi:10.4171/ifb/323 2014

  22. [22]

    Y. Chen, S. Levine, and M. Rao,Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics66(2006), no. 4, 1383–1406. DOI: 10.1137/050624522

    work page doi:10.1137/050624522 2006

  23. [23]

    M. G. Crandall, H. Ishii, and P.-L. Lions,User’s guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society (New Series)27(1992), no. 1, 1–67. DOI: 10.1090/S0273- 0979-1992-00266-5

    work page doi:10.1090/s0273- 1992

  24. [24]

    J. V. da Silva and G. C. Ricarte,Geometric gradient estimates for fully nonlinear models with non-homogeneous degeneracy and applications, Calc. Var. Partial Differential Equations59(2020), Paper No. 161. DOI: 10.1007/s00526-020-01820-7

    work page doi:10.1007/s00526-020-01820-7 2020

  25. [25]

    De Filippis,Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous degeneracy, Proc

    C. De Filippis,Regularity for solutions of fully nonlinear elliptic equations with nonhomogeneous degeneracy, Proc. Roy. Soc. Edinburgh Sect. A151(2021), no. 1, 110–132. DOI: 10.1017/prm.2020.5

    work page doi:10.1017/prm.2020.5 2021

  26. [26]

    N. M. L. Diehl and R. Teymurazyan,Reaction-diffusion equations for the infinity Laplacian, Nonlinear Anal.199 (2020), Paper No. 111956, 12 pp. DOI: 10.1016/j.na.2020.111956

    work page doi:10.1016/j.na.2020.111956 2020

  27. [27]

    dos Prazeres, R

    D. dos Prazeres, R. Teymurazyan, and J. M. Urbano,Improved regularity for a nonlocal dead-core problem, Indiana Univ. Math. J., to appear. arXiv:2504.09557

    arXiv

  28. [28]

    dos Prazeres and E

    D. dos Prazeres and E. Topp,Interior regularity results for fractional elliptic equations that degenerate with the gradient, J. Differential Equations300(2021), 814–829

    2021

  29. [29]

    Y. Fang, V. D. R˘ adulescu, and C. Zhang,Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent, Bull. Lond. Math. Soc.53(2021), no. 6, 1863–1878. DOI: 10.1112/blms.12550

    work page doi:10.1112/blms.12550 2021

  30. [30]

    Imbert and L

    C. Imbert and L. Silvestre, C 1,α regularity of solutions of some degenerate fully nonlinear elliptic equations, Adv. Math.233(2013), 196–206

    2013

  31. [31]

    F. Li, Z. Li, and L. Pi,Variable exponent functionals in image restoration, Applied Mathematics and Computation 216(2010), no. 3, 870–882. DOI: 10.1016/j.amc.2010.01.094

    work page doi:10.1016/j.amc.2010.01.094 2010

  32. [32]

    Marcon, J

    D. Marcon, J. F. Rodrigues, and R. Teymurazyan,Homogenization of obstacle problems in Orlicz–Sobolev spaces, Port. Math.75(2018), no. 3–4, 267–283

    2018

  33. [33]

    T. M. Nascimento,Schauder-type estimates for fully nonlinear degenerate elliptic equations, J. Funct. Anal.289 (2025), no. 1, Paper No. 110900, 23 pp. DOI: 10.1016/j.jfa.2025.110900

    work page doi:10.1016/j.jfa.2025.110900 2025

  34. [34]

    Nornberg, C 1,α regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, Journal de Math´ ematiques Pures et Appliqu´ ees128(2019), 297–329

    G. Nornberg, C 1,α regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, Journal de Math´ ematiques Pures et Appliqu´ ees128(2019), 297–329. DOI: 10.1016/j.matpur.2019.06.008

    work page doi:10.1016/j.matpur.2019.06.008 2019

  35. [35]

    G. C. Ricarte, J. V. da Silva, and R. Teymurazyan,Cavity type problems ruled by infinity Laplacian operator, J. Differential Equations262(2017), no. 3, 2135–2157. DOI: 10.1016/j.jde.2016.10.044

    work page doi:10.1016/j.jde.2016.10.044 2017

  36. [36]

    J. F. Rodrigues and R. Teymurazyan,On the two obstacles problem in Orlicz–Sobolev spaces and applications, Complex Var. Elliptic Equ.56(2011), no. 7–9, 769–787. DOI: 10.1080/17476933.2010.505016

    work page doi:10.1080/17476933.2010.505016 2011

  37. [37]

    E. V. Teixeira,On the critical point regularity for degenerate diffusive equations, Arch. Ration. Mech. Anal.244 (2022), no. 2, 293–316

    2022

  38. [38]

    E. V. Teixeira,Universal moduli of continuity for solutions to fully nonlinear elliptic equations, Arch. Ration. Mech. Anal.211(2014), no. 3, 911–927

    2014