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arxiv: 2605.10369 · v1 · submitted 2026-05-11 · 🧮 math.AP

Boundary C¹ regularity for degenerate fully nonlinear elliptic equations on C² domain

Jiangwen Wang , Feida Jiang This is my paper

Pith reviewed 2026-05-12 05:05 UTC · model grok-4.3

classification 🧮 math.AP
keywords degenerate elliptic equationsfully nonlinear elliptic equationsboundary regularityC^1 estimatesglobal regularityC^2 domainsHölder spaces
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The pith

Degenerate fully nonlinear elliptic equations achieve global C^1 regularity up to the boundary on C^2 domains.

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

The paper establishes that solutions to a class of degenerate fully nonlinear elliptic equations satisfy global C^{0,γ}, C^{0,1} and C^1 estimates on C^2 domains with sufficiently regular boundary data. This provides the boundary version of known interior C^1 results for the same class of equations. An explicit example demonstrates that C^{1,α} regularity on the boundary datum is optimal in the Hölder scale, and a byproduct yields global C^{1,β} estimates for certain singular fully nonlinear equations.

Core claim

We establish global regularity results (C^{0,γ}, C^{0,1} and C^1 estimates) for a class of degenerate fully nonlinear equation on C^2-domain. This corresponds to the boundary counterpart of the interior C^1 regularity results by prior works. By example we show that C^{1,α} regularity of boundary datum is sharp within the scale of Hölder spaces. As a byproduct, we also provide global C^{1,β} regularity for a class singular fully nonlinear equation.

What carries the argument

The degeneracy conditions on the fully nonlinear elliptic operator together with the C^2 smoothness of the domain, which together allow interior estimates to reach the boundary.

If this is right

  • Solutions become C^1 up to the boundary once the domain meets the C^2 threshold.
  • The same estimates apply to a corresponding class of singular equations as a byproduct.
  • C^{1,α} boundary data is the precise threshold for Hölder continuity of the gradient.
  • Global C^0,γ and C^0,1 estimates hold uniformly up to the boundary.

Where Pith is reading between the lines

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

  • The result suggests that boundary regularity for these equations is limited by domain smoothness rather than interior degeneracy alone.
  • Applications may exist in problems where degenerate equations arise on domains with limited smoothness, such as certain free-boundary or obstacle problems.
  • One could test whether the C^2 assumption on the domain can be relaxed to C^{1,1} while retaining C^1 estimates.

Load-bearing premise

The equation satisfies the specific structural degeneracy conditions that make the interior C^1 theory available, and the domain is at least C^2 with C^{1,α} boundary data.

What would settle it

An explicit solution to a degenerate fully nonlinear equation on a C^2 domain that remains merely continuous or Lipschitz but fails to be differentiable at a boundary point despite C^{1,α} boundary data.

Figures

Figures reproduced from arXiv: 2605.10369 by Feida Jiang, Jiangwen Wang.

Figure 1
Figure 1. Figure 1: Left: The 3D surface of u with a singular point (red) at (1, 0). Right: The radial profile confirming gradient blow-up. Step 1. Choose σ(t) = t 1+t , t ≥ 0, then simple calculation yields Z 1/2 0 σ −1 (s) s ds = Z 1/2 0 1 1 − s ds = ln 2 < ∞. This implies that σ −1 satisfies Dini condition. Step 2. In unit disk Ω := B1 ⊂ R 2 , let x = r sin θ, y = r cos θ, and we define u(r, θ) = uh(r, θ) + up(r), where uh… view at source ↗
read the original abstract

In this article, we establish global regularity results ($ C^{0,\gamma}$, $ C^{0,1} $ and $ C^{1}$ estimates) for a class of degenerate fully nonlinear equation on $ C^{2} $-domain. This corresponds to the boundary counterpart of the interior $ C^{1}$ regularity results by \cite{APPT22} and \cite{AN25}. By example we show that $ C^{1,\alpha} $ regularity of boundary datum is sharp within the scale of H\"{o}lder spaces. As a byproduct, we also provide global $ C^{1,\beta} $ regularity for a class singular fully nonlinear equation.

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 establishes global C^{0,γ}, C^{0,1} and C^1 regularity estimates for solutions of a class of degenerate fully nonlinear elliptic equations on C^2 domains. These results are positioned as the boundary analogue of the interior C^1 regularity theorems in APPT22 and AN25. An explicit example is given to show that C^{1,α} regularity of the boundary datum is sharp in the Hölder scale, and a byproduct is global C^{1,β} regularity for a related class of singular fully nonlinear equations.

Significance. If the proofs are complete, the work supplies a natural boundary counterpart to recent interior regularity results for degenerate fully nonlinear equations. The sharpness example is a concrete strength, as it identifies the precise Hölder threshold for the data. The byproduct for singular equations broadens the applicability. These contributions would be of interest to researchers working on boundary regularity for fully nonlinear and degenerate elliptic problems.

minor comments (3)
  1. The abstract and introduction should explicitly state the precise structural assumptions on the degeneracy (e.g., the form of the ellipticity constants or the degeneracy function) that are used to obtain the C^1 boundary estimates, rather than referring only to the interior papers.
  2. In the statement of the main boundary theorem, clarify whether the C^2 regularity of the domain is used only for the existence of a barrier or also for the construction of the boundary touching functions; a brief remark on this distinction would improve readability.
  3. The example demonstrating sharpness of C^{1,α} data should include a short verification that the constructed solution satisfies the equation in the viscosity sense up to the boundary.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. We are pleased that the work is viewed as a natural boundary counterpart to the interior C^1 regularity results of APPT22 and AN25, and that the sharpness example and byproduct for singular equations are highlighted as strengths. The recommendation for minor revision is noted; since no specific major comments were raised, we will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents global boundary regularity estimates (C^{0,γ}, C^{0,1}, C^1) for degenerate fully nonlinear equations on C^2 domains as the direct boundary analogue of interior C^1 results from the external citations APPT22 and AN25. The abstract and summary contain no self-definitional steps, no fitted parameters renamed as predictions, no load-bearing self-citations, and no ansatz or uniqueness claims that reduce to the authors' own prior unverified work. The derivation chain is self-contained against the cited external theorems and the stated hypotheses on the domain, degeneracy, and boundary data; no reduction of any load-bearing claim to the paper's own inputs by construction is present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms beyond standard PDE theory, or invented entities are mentioned in the abstract.

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