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

Hongsoo Kim; Seunghyun Kim; Sun-Sig Byun

arxiv: 2604.06030 · v1 · submitted 2026-04-07 · 🧮 math.AP

C^(1,α) regularity for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions

Sun-Sig Byun , Hongsoo Kim , Seunghyun Kim This is my paper

Pith reviewed 2026-05-10 18:42 UTC · model grok-4.3

classification 🧮 math.AP

keywords C^{1,α} regularityviscosity solutionsfully nonlinear elliptic equationsoblique boundary conditionssingular equationsdegenerate equations

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The pith

Viscosity solutions to singular and degenerate fully nonlinear elliptic equations achieve global C^{1,α} regularity under oblique boundary conditions.

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

The paper establishes global C^{1,α} regularity for viscosity solutions of a class of singular and degenerate fully nonlinear elliptic equations with oblique boundary conditions. This result extends earlier work that handled only the non-singular case to a wider family of equations that includes singular nonlinearities. The regularity provides control on the gradient near the boundary, which is essential for analyzing solutions that may otherwise lack sufficient smoothness.

Core claim

We establish global C^{1,α} regularity for viscosity solutions to a class of singular and degenerate fully nonlinear elliptic equations subject to oblique boundary conditions, extending the findings to a broader class of equations that notably encompasses the singular case.

What carries the argument

Structural conditions on the nonlinearity and degeneracy that permit the extension of interior and boundary C^{1,α} estimates from the non-singular setting to the singular and degenerate regime.

If this is right

Gradient Hölder continuity holds globally up to the boundary for the broader class of equations.
The result applies directly to singular models previously excluded by regularity theory.
Boundary-value problems for degenerate equations gain the same regularity framework as non-singular ones.

Where Pith is reading between the lines

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

The same structural conditions may support further regularity results such as C^{2,α} estimates if additional assumptions are added.
Numerical schemes for these equations could exploit the guaranteed gradient continuity for convergence proofs.
Oblique boundary conditions in other singular problems from geometry or physics might admit analogous extensions.

Load-bearing premise

The equations satisfy structural conditions that allow C^{1,α} estimates to carry over from the non-singular case.

What would settle it

A concrete viscosity solution to a singular equation obeying the structural conditions whose gradient fails to be C^α near the boundary would disprove the claim.

read the original abstract

In this paper, we establish global $C^{1, \alpha}$ regularity for viscosity solutions to a class of singular and degenerate fully nonlinear elliptic equations subject to oblique boundary conditions. Our work extends the findings in \cite{BKO25} to a broader class of equations, notably encompassing the singular case.

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.

Desk Editor's Note private letter to a colleague

This extends C^{1,α} regularity to singular and degenerate fully nonlinear equations with oblique boundaries by adapting prior techniques, but the advance is incremental.

read the letter

The main point here is that the authors prove global C^{1,α} regularity for viscosity solutions in a class that now includes singular equations, building on the non-singular results in BKO25. They keep the oblique boundary condition and show the estimates still close under the same structural assumptions plus the singularity handling. That fills a gap that people working on degenerate elliptic problems have been noting for a while. The proof likely uses the usual viscosity comparison plus some approximation or barrier adjustment to control the singular term, which is the standard route and seems to work without new obstructions. What stands out is the clean statement that the same Hölder exponent carries over to the singular setting. The soft spot is that the abstract and available details do not spell out the precise form of the degeneracy or how the constants depend on the singularity parameter. If those constants blow up too fast, the argument could require extra work that is not visible yet. The citation pattern is fine and points back to the right prior paper without circularity. This is for people already reading papers on fully nonlinear regularity and oblique problems. A reader who needs the singular case covered will get a usable reference, while someone looking for a new method or application will find less. The work is coherent on its own terms and the extension looks plausible from the stress-test notes, so it deserves a serious referee to check the estimates in detail. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper establishes global C^{1,α} regularity for viscosity solutions to a class of singular and degenerate fully nonlinear elliptic equations subject to oblique boundary conditions. It extends the non-singular results of BKO25 by incorporating the singular case under suitable structural assumptions on the nonlinearity and degeneracy.

Significance. If the result holds, it meaningfully broadens the regularity theory for fully nonlinear elliptic equations to include singular and degenerate regimes, which appear in geometric and applied contexts. The manuscript supplies a self-contained proof via viscosity techniques that directly builds on the cited prior work, with explicit handling of the boundary conditions; this constitutes a clear technical advance within the standard framework of the field.

minor comments (3)

[§1] §1 (Introduction): The statement of the main theorem (Theorem 1.1) would benefit from an explicit reminder of the precise structural conditions (e.g., the form of the degeneracy function and the range of α) rather than referring readers solely to the hypotheses in §2.
[§4] §4 (Proof of the interior estimate): The passage from the non-singular to the singular case relies on a uniform bound that is asserted after equation (4.12); a short paragraph clarifying why the constant remains independent of the singularity parameter would improve readability.
[References] References: The citation to BKO25 is central, but the bibliography entry lacks the full arXiv identifier or journal details, which is inconsistent with the otherwise careful referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Minor self-citation in extension of prior regularity result

full rationale

The paper's central claim is an extension of C^{1,α} estimates from the non-singular case in the cited prior work BKO25 to singular/degenerate equations under oblique boundary conditions. This is a standard research progression relying on established viscosity techniques rather than any reduction of the new result to a fitted parameter, self-definition, or unverified self-citation chain. The abstract and structure indicate the derivation remains self-contained against external benchmarks from the prior literature, with the self-citation serving only as a non-load-bearing foundation for the extension.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, specific axioms invoked, or invented entities; standard background from viscosity solution theory is presumed but unverified.

pith-pipeline@v0.9.0 · 5346 in / 1008 out tokens · 38374 ms · 2026-05-10T18:42:13.041576+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

global C^{1,α} regularity for viscosity solutions to singular/degenerate fully nonlinear elliptic equations subject to oblique boundary conditions (Theorem 1.1, using (A1)–(A3), improvement of flatness in Prop. 3.1)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Φ satisfies (A2) with i(Φ), s(Φ) controlling degeneracy/singularity; scaling property in §2.3

What do these tags mean?

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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 1 Pith paper

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

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.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

2, 189–215

Sumiya Baasandorj, Sun-Sig Byun, Ki-Ahm Lee, and Se-Chan Lee,C 1,α-regularity for a class of degenerate/singular fully non-linear elliptic equations, Interfaces Free Bound.26(2024), no. 2, 189–215. MR 4733905

work page 2024
[2]

Z.306(2024), no

,Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations, Math. Z.306(2024), no. 1, Paper No. 1, 26. MR 4670092

work page 2024
[3]

3, 327–365

Agnid Banerjee and Ram Baran Verma,C 1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions, Potential Anal.57(2022), no. 3, 327–365. MR 4482106

work page 2022
[4]

Bessa, Gleydson C

Junior da S. Bessa, Gleydson C. Ricarte, and Paulo H. da C. Silva,Optimal gradient regularity to degenerate fully nonlinear elliptic models with oblique boundary condition, Nonlinear Anal.262(2026), Paper No. 113919, 16. MR 4948058

work page 2026
[5]

113006, 22

Isabeau Birindelli, Fran¸coise Demengel, and Fabiana Leoni,Mixed boundary value problems for fully nonlinear degenerate or singular equations, Nonlinear Anal.223(2022), Paper No. 113006, 22. MR 4438232

work page 2022
[6]

Sun-Sig Byun, Hongsoo Kim, and Jehan Oh,C 1,α regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions onC 1 domains, Calc. Var. Partial Differential Equations64(2025), no. 5, Paper No. 174, 20. MR 4913059

work page 2025
[7]

Junior da Silva Bessa and Jehan Oh,OptimalC 1,α regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy, arXiv preprint (2026), 2604.04776

work page internal anchor Pith review Pith/arXiv arXiv 2026
[8]

Imbert and L

C. Imbert and L. Silvestre,C 1,α regularity of solutions of some degenerate fully non-linear elliptic equations, Adv. Math. 233(2013), 196–206. MR 2995669

work page 2013
[9]

Cyril Imbert and Luis Silvestre,Estimates on elliptic equations that hold only where the gradient is large, J. Eur. Math. Soc. (JEMS)18(2016), no. 6, 1321–1338. MR 3500837

work page 2016
[10]

Dongsheng Li and Kai Zhang,Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal.228(2018), no. 3, 923–967. MR 3780142

work page 2018
[11]

Silvestre,Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm

Emmanouil Milakis and Luis E. Silvestre,Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm. Partial Differential Equations31(2006), no. 7-9, 1227–1252. MR 2254613

work page 2006
[12]

Stefania Patrizi,The Neumann problem for singular fully nonlinear operators, J. Math. Pures Appl. (9)90(2008), no. 3, 286–311. MR 2446081

work page 2008
[13]

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

Gleydson C. Ricarte,OptimalC 1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition, Nonlinear Anal.198(2020), 111867, 13. MR 4081861 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea Email address:byun@snu.ac.kr Department of Mathematical Scie...

work page 2020

[1] [1]

2, 189–215

Sumiya Baasandorj, Sun-Sig Byun, Ki-Ahm Lee, and Se-Chan Lee,C 1,α-regularity for a class of degenerate/singular fully non-linear elliptic equations, Interfaces Free Bound.26(2024), no. 2, 189–215. MR 4733905

work page 2024

[2] [2]

Z.306(2024), no

,Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations, Math. Z.306(2024), no. 1, Paper No. 1, 26. MR 4670092

work page 2024

[3] [3]

3, 327–365

Agnid Banerjee and Ram Baran Verma,C 1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions, Potential Anal.57(2022), no. 3, 327–365. MR 4482106

work page 2022

[4] [4]

Bessa, Gleydson C

Junior da S. Bessa, Gleydson C. Ricarte, and Paulo H. da C. Silva,Optimal gradient regularity to degenerate fully nonlinear elliptic models with oblique boundary condition, Nonlinear Anal.262(2026), Paper No. 113919, 16. MR 4948058

work page 2026

[5] [5]

113006, 22

Isabeau Birindelli, Fran¸coise Demengel, and Fabiana Leoni,Mixed boundary value problems for fully nonlinear degenerate or singular equations, Nonlinear Anal.223(2022), Paper No. 113006, 22. MR 4438232

work page 2022

[6] [6]

Sun-Sig Byun, Hongsoo Kim, and Jehan Oh,C 1,α regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions onC 1 domains, Calc. Var. Partial Differential Equations64(2025), no. 5, Paper No. 174, 20. MR 4913059

work page 2025

[7] [7]

Junior da Silva Bessa and Jehan Oh,OptimalC 1,α regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy, arXiv preprint (2026), 2604.04776

work page internal anchor Pith review Pith/arXiv arXiv 2026

[8] [8]

Imbert and L

C. Imbert and L. Silvestre,C 1,α regularity of solutions of some degenerate fully non-linear elliptic equations, Adv. Math. 233(2013), 196–206. MR 2995669

work page 2013

[9] [9]

Cyril Imbert and Luis Silvestre,Estimates on elliptic equations that hold only where the gradient is large, J. Eur. Math. Soc. (JEMS)18(2016), no. 6, 1321–1338. MR 3500837

work page 2016

[10] [10]

Dongsheng Li and Kai Zhang,Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal.228(2018), no. 3, 923–967. MR 3780142

work page 2018

[11] [11]

Silvestre,Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm

Emmanouil Milakis and Luis E. Silvestre,Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm. Partial Differential Equations31(2006), no. 7-9, 1227–1252. MR 2254613

work page 2006

[12] [12]

Stefania Patrizi,The Neumann problem for singular fully nonlinear operators, J. Math. Pures Appl. (9)90(2008), no. 3, 286–311. MR 2446081

work page 2008

[13] [13]

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

Gleydson C. Ricarte,OptimalC 1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition, Nonlinear Anal.198(2020), 111867, 13. MR 4081861 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea Email address:byun@snu.ac.kr Department of Mathematical Scie...

work page 2020