Interior $C^{2}$ estimate for semi-convex solutions to a class of Hessian quotient equations in arbitrary dimensions

Abstract: In this paper, we study the interior $C^{2}$ estimates for Hessian quotient equations $\frac{σ{3}(D^{2}u)}{σ{l}(D^{2}u)}=1$ for $l=1, 2$, in arbitrary dimensions, under the natural ellipticity and semi-convexity conditions. We further derive analogous results for the corresponding sum Hessian equations. In addition, we establish several rigidity results.

• The paper establishes interior C² estimates for semi-convex solutions to Hessian quotient equations with explicit bounds dependent on geometry and regularity norms.
• It develops a multi-step analytic strategy that bypasses traditional concavity constraints and proves sharp rigidity theorems for sub-quadratic growth solutions.
• The work constructs singular solutions for higher-order cases, confirming the precise threshold for regularity in fully nonlinear elliptic PDEs.

Interior $C^{2}$ Estimates for Semi-Convex Solutions to Hessian Quotient Equations in Arbitrary Dimensions

Introduction and Context

This work (2604.23349) resolves a longstanding question in the theory of fully nonlinear elliptic PDEs regarding interior regularity for semi-convex solutions of certain Hessian quotient equations in arbitrary dimensions. Specifically, it establishes interior $C^{2}$ estimates for semi-convex and admissible solutions to the equations

$\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$

for $l=1,2$, where $\sigma_k$ denotes the $k$-th elementary symmetric function of the eigenvalues of $D^2 u$. It further extends the results to sum Hessian equations and provides new rigidity theorems.

Main Results

Interior $C^{2}$ Estimates

The principal result asserts that for $n\geq 3$, if $u\in C^4(B_5)$ is semi-convex and admissible (i.e., $C^{2}$0), then: $C^{2}$1 where $C^{2}$2 depends only on $C^{2}$3, the semi-convexity constant $C^{2}$4, and $C^{2}$5. This holds for the Hessian quotient equations with $C^{2}$6, providing the first affirmative answer for these cases in arbitrary dimensions without recourse to special geometric structures (such as those present in low-dimensional special Lagrangian equations).

Additionally, analogous $C^{2}$7 estimates are demonstrated for sum Hessian equations $C^{2}$8, where $C^{2}$9 and $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$0 is positive and regular, under the same admissibility and semi-convexity condition.

Rigidity Theorems

A direct corollary is a rigidity result: any entire admissible semi-convex solution with sub-quadratic growth to the quotient equation must be a quadratic polynomial. Similar rigidity theorems are established for the sum Hessian equations under varying convexity and growth conditions, including removal of the quadratic growth assumption for convex solutions.

Singular Solutions

For higher-order equations ($\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$1), the paper explicitly constructs singular solutions, thereby illustrating that interior $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$2 estimates fail in those cases. The provided examples generalize Pogorelov's constructions, showing that the regularity threshold is sharp for these classes.

Technical Approach

The work innovates by circumventing the lack of standard concavity inequalities for Hessian quotient operators when $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$3. Instead, it leverages results on the ellipticity and concavity of sum Hessian operators $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$4 in the intersection cone $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$5, utilizing results by Li-Ren-Wang for ellipticity.

A multi-step proof strategy is employed:

• Step 1: Establishes a Jacobi-type inequality for $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$6, relying on a higher-dimensional concavity inequality for $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$7 derived through recent results.
• Step 2: Uses the Legendre transform and a mean-value type inequality to relate the pointwise value $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$8 to an integral over a neighborhood, by proving uniform ellipticity for the transformed operator.
• Step 3: Manages integral terms via integration by parts and careful comparison of key quantities $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)} = 1$9 and $l=1,2$0, drawing on a technical lemma restricting their relationship.

The strategy is designed to handle the inherent difficulties posed by the non-homogeneous nature of sum Hessian equations and the fact that boundedness of certain eigenvalues is not available.

Numerical and Structural Highlights

• The $l=1,2$1 bound is explicit and depends only on parameters directly associated with the solution's geometry and prescribed regularity.
• The rigidity theorem for sub-quadratic growth solutions is sharp and excludes non-trivial entire solutions outside the quadratic class.
• Construction of singular solutions for $l=1,2$2 is rigorous and demonstrates the boundary of possible $l=1,2$3 regularity.
• The arguments fully generalize previously dimension-limited results, such as those of Warren–Yuan and Shankar–Yuan, to arbitrary dimensions.

Implications and Future Directions

Practically, these results widen the scope of regularity theory for nonlinear PDEs governing geometric structures and generalize classical interior estimates for Monge-Ampère and $l=1,2$4-Hessian equations into more exotic settings.

Theoretically, the concavity and ellipticity properties elucidated for sum Hessian operators suggest robustness in the structure of certain fully nonlinear elliptic PDEs, even in the absence of explicit geometric interpretations.

Remaining open problems include extending $l=1,2$5 estimates to general sum Hessian equations for arbitrary admissible solutions without convexity assumptions, and further classification of entire solutions outside the quadratic class for larger $l=1,2$6.

While not directly related to computational AI, advances in PDE regularity—especially in high-dimensional contexts—can influence numerical solvers, geometric optimization, and potential functional analysis of neural network architectures, especially where Hessian-based loss functions or constraints arise.

Conclusion

This paper establishes new, dimension-agnostic interior $l=1,2$7 estimates for semi-convex solutions of Hessian quotient and sum Hessian equations and rigorously proves rigidity results, expanding the boundary of classical nonlinear elliptic regularity theory. The techniques developed pave the way for future investigations into higher-order nonlinear PDEs, their singularities, and classification of solution behavior in geometric and analytic settings. The sharp identification of the regularity threshold via construction of singular solutions is an essential contribution to the theory.