pith. sign in

arxiv: 2604.04776 · v1 · submitted 2026-04-06 · 🧮 math.AP

Optimal C^(1,α) regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy

Junior da Silva Bessa , Jehan Oh This is my paper

Pith reviewed 2026-05-10 19:53 UTC · model grok-4.3

classification 🧮 math.AP
keywords fully nonlinear elliptic equationsdouble phase degeneracyviscosity solutionsboundary regularityC^{1,alpha} regularityoblique boundary conditionsHölder estimates
0
0 comments X

The pith

Viscosity solutions to fully nonlinear elliptic equations with double phase degeneracy attain optimal C^{1,α} regularity up to the boundary under oblique conditions.

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

The paper establishes optimal C^{1,α} regularity up to the boundary for viscosity solutions to fully nonlinear elliptic equations that feature a double phase degeneracy law, subject to oblique boundary conditions. This is achieved by first deriving uniform boundary Hölder estimates for perturbed models in almost C¹-flat domains with oblique boundary data. Then, a compactness and stability argument for viscosity solutions yields the desired regularity result. As a result, the optimal Hölder exponent is identified when the operator is quasiconvex or quasiconcave, and regularity improves at points where the source term vanishes.

Core claim

Viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions satisfy optimal C^{1,α} regularity up to the boundary in almost C¹-flat domains. The proof first obtains uniform boundary Hölder estimates on perturbed models and then passes to the limit using compactness and stability in the viscosity framework. This determines the optimal exponent for quasiconvex or quasiconcave operators and gives improved regularity along vanishing points of the source term.

What carries the argument

Uniform boundary Hölder estimates on perturbed models in almost C¹-flat domains, followed by a compactness-stability framework for viscosity solutions.

If this is right

  • Viscosity solutions achieve the optimal Hölder exponent α for quasiconvex or quasiconcave governing operators.
  • Improved regularity holds along points where the source term vanishes.
  • The regularity result extends to oblique boundary conditions.
  • The approach applies in domains that are almost C¹-flat.

Where Pith is reading between the lines

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

  • These boundary estimates might combine with known interior regularity to give global C^{1,α} results in suitable domains.
  • Similar techniques could apply to other types of degeneracy or boundary conditions in fully nonlinear problems.
  • The stability framework may extend to settings with lower regularity on the coefficients.

Load-bearing premise

The domains must be almost C¹-flat and the double phase degeneracy must satisfy structural conditions that permit uniform boundary Hölder estimates on the perturbed models.

What would settle it

Finding a viscosity solution to such an equation in an almost C¹-flat domain with double phase degeneracy that fails to be differentiable with Hölder continuous gradient at some boundary point would disprove the claim.

read the original abstract

In this paper we establish optimal $C^{1,\alpha}$ regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here relies on first deriving uniform boundary H\"older estimates for perturbed models with oblique boundary data in ``almost $C^{1}$-flat'' domains. Building upon these estimates, the desired regularity is obtained through a compactness and stability framework for viscosity solutions. As a byproduct of our analysis, we determine the optimal H\"older exponent for solutions when the governing operator is quasiconvex or quasiconcave. In addition, we establish an improved regularity result along vanishing points of the source term.

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 / 3 minor

Summary. The paper claims to establish optimal C^{1,α} regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy and oblique boundary conditions. The proof proceeds in two steps: first deriving uniform boundary Hölder estimates for perturbed models in almost C¹-flat domains, then passing to the limit via a compactness-stability argument in the viscosity framework. Byproducts include the optimal Hölder exponent in the quasiconvex/quasiconcave case and improved regularity at vanishing points of the source term.

Significance. If the central claims hold, the work meaningfully extends boundary regularity theory for degenerate fully nonlinear equations, handling the double-phase structure and oblique conditions where interior results are more common. The optimality statement and the byproduct on vanishing points are concrete strengths; the perturbed-model-plus-stability approach is a standard, reproducible technique in the viscosity literature and, when carried out carefully, yields falsifiable predictions for the exponent α.

minor comments (3)
  1. The precise structural assumptions on the double-phase degeneracy (e.g., the range of the exponents and the ellipticity constants) should be stated explicitly in the introduction or §2 rather than deferred entirely to the equation list, to make the optimality claim immediately verifiable.
  2. The definition and quantitative control of “almost C¹-flat” domains (central to the boundary Hölder estimates) needs a self-contained paragraph or display in the preliminaries; without it, readers cannot immediately check how the flatness parameter enters the constants in the first step.
  3. A short comparison paragraph with existing single-phase boundary results (e.g., those using similar compactness arguments) would clarify the novelty of the double-phase passage and the role of the oblique condition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The referee's summary accurately reflects the main results and approach of the manuscript. Since no specific major comments are listed, we have no points requiring rebuttal or revision at this time.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's approach first obtains uniform boundary Hölder estimates on perturbed models with oblique data in almost C¹-flat domains, then passes to the limit via compactness and stability in the viscosity solution framework. These steps rely on independent structural assumptions on the double-phase degeneracy (ellipticity, growth) and domain flatness, without reducing the target C^{1,α} regularity to a fitted parameter, self-definition, or load-bearing self-citation. The byproduct optimal exponent for quasiconvex/quasiconcave cases and improved regularity at vanishing points follow from the same estimates and do not presuppose the final result. This matches standard non-circular techniques in fully nonlinear regularity theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract provides no explicit free parameters or invented entities; relies on standard viscosity solution axioms and domain flattening assumptions.

axioms (2)
  • domain assumption Solutions are understood in the viscosity sense for fully nonlinear operators
    Standard framework invoked for the equations and boundary conditions.
  • domain assumption Domains admit almost C¹-flat approximations
    Used to derive uniform boundary Hölder estimates for perturbed models.

pith-pipeline@v0.9.0 · 5415 in / 1307 out tokens · 40971 ms · 2026-05-10T19:53:08.192850+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

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

  1. Sharp regularity for degenerate fully nonlinear equations with oblique boundary conditions and Hamiltonian terms

    math.AP 2026-05 unverdicted novelty 7.0

    Optimal boundary C^{1,α} regularity is proved for viscosity solutions to degenerate fully nonlinear equations with oblique boundary conditions and Hamiltonian terms.

  2. C^{1,\alpha} regularity for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions

    math.AP 2026-04 unverdicted novelty 5.0

    Global C^{1,α} regularity is established for viscosity solutions of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions, extending previous work to the singular case.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 2 Pith papers

  1. [1]

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

    work page 2015

  2. [2]

    D. J. Ara´ ujo and L. Zhang, OptimalC1,α estimates for a class of elliptic quasilinear equations, Comm. Contemp. Math.22(2020), no. 5, 1950062

    work page 2020

  3. [3]

    Banerjee and R

    A. Banerjee and R. B. Verma,C 1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions, Potential Anal.57(2022), no. 3, 327–365

    work page 2022

  4. [4]

    Barles, Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations, J

    G. Barles, Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations, J. Differential Equations106(1993), no. 1, 90–106

    work page 1993

  5. [5]

    Baroni, M

    P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal.121(2015), 206–222

    work page 2015

  6. [6]

    J. da S. Bessa, J. V. da Silva and G. C. Ricarte, Sharp moduli of continuity for solutions to fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations 455(2026), Paper No. 113961, 42 pp

    work page 2026

  7. [7]

    J. da S. Bessa, G. C. Ricarte and P. H. C. Silva, Optimal gradient regularity to degenerate fully nonlinear elliptic models with oblique boundary condition, Nonlinear Anal.262(2026), Paper No. 113919, 16 pp

    work page 2026

  8. [8]

    Birindelli and F

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

    work page 2004

  9. [9]

    Birindelli and F

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

    work page 2007

  10. [10]

    Birindelli, F

    I. Birindelli, F. Demengel and F. Leoni,C 1,γ regularity for singular or degenerate fully non- linear equations and applications, NoDEA Nonlinear Differential Equations Appl.26(2019), no. 5, Art. 40

    work page 2019

  11. [11]

    S.-S. Byun, H. Kim and J. Oh,C 1,α regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions onC 1 domains, Calc. Var. Partial Differential Equations 64(2025), no. 5, Paper No. 174, 20 pp

    work page 2025

  12. [12]

    Colombo and G

    M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal.215(2015), no. 2, 443–496

    work page 2015

  13. [13]

    Colombo and G

    M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal.218(2015), no. 1, 219–273

    work page 2015

  14. [14]

    Colombo and G

    M. Colombo and G. Mingione, Calder´ on–Zygmund estimates and non-uniformly elliptic op- erators, J. Funct. Anal.270(2016), no. 4, 1416–1478

    work page 2016

  15. [15]

    small ellipticity aperture

    J. V. da Silva and M. S. Santos, Schauder and Calder´ on-Zygmund type estimates for fully nonlinear parabolic equations under “small ellipticity aperture” and applications, Nonlinear Anal.246(2024), Paper No. 113578, 17 pp

    work page 2024

  16. [16]

    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), no. 5, Paper No. 161, 33 pp

    work page 2020

  17. [17]

    De Filippis, Gradient bounds for solutions to irregular parabolic equations with (p, q)- growth, Calc

    C. De Filippis, Gradient bounds for solutions to irregular parabolic equations with (p, q)- growth, Calc. Var. Partial Differential Equations59(2020), no. 5, Paper No. 171, 32 pp. 17

    work page 2020

  18. [18]

    De Filippis, Regularity for solutions of fully nonlinear elliptic equations with nonhomoge- neous degeneracy, Proc

    C. De Filippis, Regularity for solutions of fully nonlinear elliptic equations with nonhomoge- neous degeneracy, Proc. Roy. Soc. Edinburgh Sect. A151(2021), no. 1, 110–132

    work page 2021

  19. [19]

    De Filippis and G

    C. De Filippis and G. Mingione, Nonuniformly elliptic Schauder theory, Invent. Math.234 (2023), no. 3, 1109–1196

    work page 2023

  20. [20]

    De Filippis and G

    C. De Filippis and G. Mingione, The sharp growth rate in nonuniformly elliptic Schauder theory, Duke Math. J.174(2025), no. 9, 1775–1848

    work page 2025

  21. [21]

    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

    work page 2021

  22. [22]

    Imbert and L

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

    work page 2013

  23. [23]

    Ishii, Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDE’s, Duke Math

    H. Ishii, Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDE’s, Duke Math. J.62(1991), no. 3, 633–661

    work page 1991

  24. [24]

    D. S. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal.228(2018), no. 3, 923–967

    work page 2018

  25. [25]

    Lions and N

    P.-L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic Hamilton–Jacobi–Bellman equation, Math. Z.191(1986), no. 1, 1–15

    work page 1986

  26. [26]

    Mingione and V

    G. Mingione and V. D. R˘ adulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl.501(2021), no. 1, Paper No. 125197

    work page 2021

  27. [27]

    Milakis and L

    E. Milakis and L. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm. Partial Differential Equations31(2006), no. 7–9, 1227–1252

    work page 2006

  28. [28]

    Nadirashvili and S

    N. Nadirashvili and S. Vl˘ adut ¸, Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal.17(2007), no. 4, 1283–1296

    work page 2007

  29. [29]

    Nadirashvili and S

    N. Nadirashvili and S. Vl˘ adut ¸,Singular viscosity solutions to fully nonlinear elliptic equations. J. Math. Pures Appl. (9) 89 (2) (2008) 107–113

    work page 2008

  30. [30]

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

    work page 2025

  31. [31]

    G. C. Ricarte, OptimalC 1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition, Nonlinear Anal.198(2020), 111867, 13 pp

    work page 2020

  32. [32]

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

    work page 2014

  33. [33]

    E. V. Teixeira, Regularity for quasilinear equations on degenerate singular sets, Math. Ann. 358(2014), no. 1–2, 241–256

    work page 2014

  34. [34]

    Wu and P

    D. Wu and P. Niu, Interior pointwiseC 2,α regularity for fully nonlinear elliptic equations, Nonlinear Anal.227(2023), Paper No. 113159, 9 pp

    work page 2023

  35. [35]

    V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat.50(1986), no. 4, 675–710; English transl., Math. USSR-Izv.29 (1987), no. 1, 33–66

    work page 1986

  36. [36]

    V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys.3(1995), no. 2, 249–269. 18 Junior da Silva Bessa Universidade Estadual de Campinas Instituto de Matem´ atica, Estat´ ıstica e Computa¸ c˜ ao Cient´ ıfica - IMECC Departamento de Matem´ atica Rua S´ ergio Buarque de Holanda, 651 Campinas - SP, Brazil 13083-859 jbessa@unicamp.br Jehan Oh Kyung...

    work page 1995