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arxiv: 2604.09044 · v1 · submitted 2026-04-10 · 🧮 math.AP

The Neumann problem for a class of degenerate Hessian quotient type equations

Jiabao Gong , Qiang Tu This is my paper

Pith reviewed 2026-05-10 17:15 UTC · model grok-4.3

classification 🧮 math.AP
keywords Hessian quotient operatorsNeumann problemdegenerate elliptic equationsa priori estimatesexistence theoremfully nonlinear PDEelementary symmetric functions
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The pith

Inequalities for generalized Hessian quotient operators enable global estimates and existence for degenerate Neumann problems with extended k.

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

The paper derives key inequalities for operators of the form σ_k(Λ(D²u))/σ_l(Λ(D²u)), viewed as a generalization of classical Hessian quotients. These inequalities underpin global a priori estimates for solutions of the associated degenerate equations. The estimates in turn yield an existence theorem for the Neumann problem when k satisfies 0 < k ≤ C_n^p with 1 ≤ p ≤ n-1. A reader cares because the result enlarges the class of fully nonlinear boundary-value problems that can be treated by a priori methods.

Core claim

We obtain some important inequalities for a class of Hessian quotient type operators σ_k(Λ(D²u))/σ_l(Λ(D²u)), which can be regarded as a generalization of the classical Hessian quotient operators. As an application, we establish global a priori estimates and prove an existence theorem for the Neumann problem of the corresponding degenerate Hessian quotient type equation, in which the admissible range of k is extended to 0 < k ≤ C^p_n with 1 ≤ p ≤ n-1.

What carries the argument

The inequalities obtained for the generalized Hessian quotient type operators σ_k(Λ(D²u))/σ_l(Λ(D²u)), which serve as the foundation for the global estimates.

If this is right

  • Global a priori estimates hold for solutions of the Neumann problem under the extended range of k.
  • An existence theorem is obtained for the degenerate Hessian quotient type equation.
  • The admissible values of k now include the full interval up to the binomial coefficient C_n^p.

Where Pith is reading between the lines

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

  • The same inequalities may support existence results under Dirichlet or oblique boundary conditions for related equations.
  • The technique could be tested numerically on low-dimensional cases to check whether the bounds on k are sharp.
  • The operators might be useful in studying curvature flows or other geometric PDEs that involve ratios of symmetric functions.

Load-bearing premise

The eigenvalues of the Hessian must satisfy the admissibility conditions that make the quotient operators positive and elliptic, and the domain together with the boundary data must be smooth enough for the estimates to close.

What would settle it

A concrete smooth bounded domain, smooth boundary function, and smooth right-hand side for which the Neumann problem admits no solution when k lies in the claimed extended interval 0 < k ≤ C_n^p for some p.

read the original abstract

In this paper, we obtain some important inequalities for a class of Hessian quotient type operators $\frac{\sigma_k(\Lambda(D^2u))}{\sigma_l(\Lambda(D^2u))}$, which can be regarded as a generalization of the classical Hessian quotient operators. As an application, we establish global a priori estimates and prove an existence theorem for the Neumann problem of the corresponding degenerate Hessian quotient type equation, in which the admissible range of $k$ is extended to $0< k \leq C^\mathbf{p}_n$ with $1 \leq \mathbf{p} \leq n-1$.

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 derives some important inequalities for the generalized Hessian quotient type operators given by the ratio σ_k(Λ(D²u))/σ_l(Λ(D²u)), viewed as an extension of classical Hessian quotients. Using these, the authors establish global a priori estimates and prove existence for the Neumann problem of the associated degenerate equation, with the range of admissible k extended to real numbers satisfying 0 < k ≤ binom(n,p) for 1 ≤ p ≤ n-1.

Significance. Assuming the inequalities are correctly established under the admissibility cone conditions, this work meaningfully extends the theory of fully nonlinear degenerate elliptic equations to non-integer parameters k in the quotient operators. The global estimates for the Neumann problem are obtained via the continuity method, with the algebraic inequalities controlling the necessary terms. This generalization broadens the applicability to more general curvature-type problems. Strengths include the direct derivation of the inequalities without hidden parameters and the handling of degeneracy within the open cone.

minor comments (3)
  1. Clarify the notation C^p_n in the abstract and throughout; it appears to denote the binomial coefficient binom(n,p).
  2. Add more references to prior works on Hessian quotient operators and Neumann problems for fully nonlinear equations to better situate the contribution.
  3. In the section deriving the a priori estimates, provide additional detail on how the boundary terms are controlled under the Neumann condition when the operator is degenerate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of our results on generalized Hessian quotient inequalities, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper first establishes algebraic inequalities for the generalized quotient operators σ_k(Λ(D²u))/σ_l(Λ(D²u)) under the stated cone conditions on the eigenvalues of the Hessian. These inequalities are derived from properties of elementary symmetric polynomials and are independent of the target PDE solution. They are then applied to control second derivatives and boundary terms in the continuity method for the Neumann problem, yielding global a priori estimates and existence. The extension of the admissible range for k to (0, C^p_n] follows directly from the algebraic inequalities without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. The argument is self-contained against standard fully nonlinear PDE techniques and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain and boundary smoothness assumptions typical for elliptic PDE existence theorems, plus admissibility conditions on the eigenvalues of the Hessian; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The domain is smooth and the boundary data satisfy compatibility conditions for the Neumann problem.
    Standard prerequisite for global a priori estimates in boundary value problems for elliptic equations.
  • domain assumption The eigenvalues lie in the admissible cone where the quotient operator is elliptic and positive.
    Required for the inequalities and estimates to hold as stated in the abstract.

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Cited by 1 Pith paper

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

  1. The existence of $(\mathbf{p}, k)$-convex hypersurfaces for a class of Hessian quotient type curvature equations

    math.AP 2026-04 unverdicted novelty 5.0

    Existence and uniqueness of (p,k)-convex hypersurfaces are established for Hessian quotient type curvature equations via a priori estimates, continuity method, and an inverse convexity property yielding a constant ran...

Reference graph

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