Pogorelov type interior $C^2$ estimate for Hessian quotient equation and its application

TL;DR

The paper proves a Pogorelov-type interior $C^2$ estimate for the Hessian quotient equation $σ_n/σ_k(D^2u)=f$, enabling interior regularity results for convex viscosity solutions in the regime $k≤n-3$. The authors combine a Legendre-transform approach, concavity properties of the Hessian and Hessian quotient operators, Jacobi-type inequalities, and mean-value/elliptic-regularity techniques to derive a uniform interior bound on the largest eigenvalue and hence on $D^2u$. This yields $C^{3,β}$-type interior regularity for convex viscosity solutions under sharp conditions on $α$ or $p$ (the exponents are shown to be sharp), and extends the regularity theory from Monge–Ampère and Hessian equations to Hessian quotients. The results have implications for the regularity and convexity of solutions to fully nonlinear Hessian-quotient equations under weak boundary-vanishing and convexity assumptions, with a robust framework for further generalizations.

Abstract

In this paper, we derive a Pogorelov type interior $C^2$ estimate for the Hessian quotient equation $\frac{σ_n}{σ_k}\left( D^2u\right) =f$. As an application, we show that convex viscosity solutions are regular for $k\leq n-3$ if $u\in C^{1,α}$ with $α>1-\frac{2}{n-k}$ or $u\in W^{2,p}$ with $p\geq\frac{(n-1)(n-k)}{2}$. Both exponents are sharp in view of the example in arXiv:2401.12229.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1: Zhang RZhang