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

Abstract: In this paper, we obtain some important inequalities for a class of Hessian quotient type operators $\frac{σ_k(Λ(D^2u))}{σ_l(Λ(D^2u))}$, 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 $1 \leq \mathbf{p} \leq n-1$.

• The paper establishes existence and uniqueness of weak solutions for the degenerate Hessian quotient Neumann problem under relaxed ellipticity conditions.
• It derives sharp derivative and boundary estimates using innovative auxiliary functions and precise comparison inequalities.
• Utilizing regularization and maximum principle techniques, the work unifies degenerate and non-degenerate cases, extending classical PDE results.

The Neumann Problem for Degenerate Hessian Quotient Type Equations

Introduction and Context

The addressed work systematically investigates the Neumann boundary value problem for a class of degenerate Hessian quotient type equations involving operators of the form

$$F(D^2 u) = \frac{\sigma_k(\Lambda(D^2 u))}{\sigma_l(\Lambda(D^2 u))} = f(x) \quad \text{in } \Omega,$$

where $\Omega \subset \mathbb{R}^n$ is a strictly convex domain with $C^4$ boundary, $\sigma_k$ is the $k$-th elementary symmetric function, and $\Lambda(D^2 u)$ encodes sums of eigenvalues of the Hessian $D^2 u$ taken over multi-indices of size $\mathbf{p}$. The admissibility condition on $u$ is determined via the Gårding cone, adapted to this generalized setting.

Hessian quotient equations and their degenerate, fully nonlinear counterparts underpin several geometric PDEs, notably including real and complex Monge-Ampère, $k$-Hessian, and form-type Calabi-Yau or Gauduchon-type equations. This paper extends the reach of existing methods to a broader parameter range $\Omega \subset \mathbb{R}^n$0 with $\Omega \subset \mathbb{R}^n$1, relaxing previous ellipticity constraints and encompassing new degenerate scenarios.

Main Results

Foundational Inequalities and Operator Properties

A central technical advancement is the derivation of sharp derivative estimates and concavity properties for the generalized Hessian quotient operator $\Omega \subset \mathbb{R}^n$2. The paper proves comparison properties analogous to those previously known only in the strictly elliptic regime ($\Omega \subset \mathbb{R}^n$3), and shows these extend under milder restrictions to cases where $\Omega \subset \mathbb{R}^n$4 reaches the maximal combinatorial value $\Omega \subset \mathbb{R}^n$5. Key lemmas support the establishment of uniform gradient and second derivative estimates, including strong lower bounds for directional derivatives and precise behavior under variations of the eigenstructure, even when the operator degenerates.

A Priori Estimates and Regularity

The analysis employs refined maximum principle and test function arguments, inspired by earlier work of Lions-Trudinger-Urbas on fully nonlinear equations. The authors construct auxiliary functions suited to the exterior algebraic setting, enabling the control of double normal derivatives and boundary behavior. For positive data $\Omega \subset \mathbb{R}^n$6 with $\Omega \subset \mathbb{R}^n$7, solutions are shown to enjoy uniform $\Omega \subset \mathbb{R}^n$8 regularity when degeneracy is allowed, and $\Omega \subset \mathbb{R}^n$9 regularity in the non-degenerate case.

Existence and Uniqueness of Weak Solutions

Utilizing the established a priori bounds, existence and uniqueness of $C^4$0-admissible weak solutions to the Neumann boundary problem are obtained via approximating the degenerate equation by a family of perturbed Neumann problems. By passing to the limit in the regularization parameter and employing compactness arguments, a unique weak solution in $C^4$1 is obtained, unique up to an additive constant due to the invariance of the Neumann problem under vertical shifts.

This result, formulated precisely in Theorem 1, significantly extends the previously known existence theorems to the degenerate ($C^4$2) and maximal parameter regime. For strictly positive $C^4$3, the classical method of continuity yields classical solutions in higher regularity spaces.

Analytical Contributions

Several technical contributions underpin these results:

• Derivative comparison inequalities: The paper demonstrates that the directional derivatives of the operator with respect to the eigenvalues satisfy lower bounds on each cone slice, extending key tools from symmetric function theory to the generalized setting.
• Concavity and monotonicity: It is proved that $C^4$4 retains concavity and monotonicity on the enlarged admissible domain $C^4$5.
• Boundary double normal estimate: A subtle auxiliary function construction yields uniform boundary estimates for $C^4$6, even absent a uniform lower bound on $C^4$7.
• Limiting procedure and uniqueness: The approximation scheme, following the Lions-Trudinger-Urbas template, constructs solutions to the degenerate problem as limits of uniformly elliptic problems with penalized Neumann data, carefully tracking constants to ensure admissibility and uniqueness are preserved.
• Unified treatment of degenerate and non-degenerate cases: The methodology provides a transparent passage between degenerate and strictly elliptic cases, offering a template for parallel results on manifolds or other nonlinear equations involving combinations of traces and symmetric functions.

Implications and Future Directions

The theoretical framework developed here strengthens the analytic foundation for degenerate fully nonlinear elliptic equations of Hessian quotient type. By broadening the admissible range for critical parameters, it opens the door for further geometric and analytic applications, including form-type PDEs on complex manifolds, degenerate prescribing curvature problems, and generalizations to Riemannian or Hermitian settings.

From a PDE and geometric analysis perspective, the sharp a priori estimates and strong-weak solution theory facilitate more refined regularity and stability analyses beyond the convex Euclidean case. There are direct implications for the study of geometric flows, degenerate complex Monge-Ampère equations, and equations of mixed order arising in calibrated geometry.

Future work may pursue extension to systems, lower regularity domains, measure data, or explore nonlocal analogues. In the context of geometric analysis, deeper exploration of the interplay between degeneracy, geometric structure, and boundary regularity remains a rich direction.

Conclusion

This work provides comprehensive existence, uniqueness, and regularity results for the Neumann problem of a wide class of degenerate Hessian quotient equations, extending central tools of modern nonlinear elliptic PDE theory to encompass degenerate and maximally non-elliptic regimes. The paper's methodological advances, particularly regarding operator inequalities and a priori boundary estimates, set a robust precedent for further explorations of degenerate fully nonlinear problems in both real and complex domains (2604.09044).