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

TL;DR

This paper establishes a Pogorelov type interior $C^2$ estimate for the Hessian quotient equation and demonstrates the regularity of convex viscosity solutions under certain conditions, extending the understanding of solution smoothness.

Contribution

It introduces a new interior $C^2$ estimate for the Hessian quotient equation and applies it to prove regularity of convex solutions in specific cases.

Findings

Convex viscosity solutions are regular for $k extless n-2$ under certain smoothness conditions.

The derived estimates are sharp, matching known examples.

The results extend regularity theory for Hessian quotient equations.

Abstract

In this paper, we derive a Pogorelov type interior $C^2$ estimate for the Hessian quotient equation $\frac{\sigma _n}{\sigma _k}\left( D^2u\right) =f$. As an application, we show that convex viscosity solutions are regular for $k\leq n-3$ if $u\in C^{1,\alpha}$ with $\alpha>1-\frac{2}{n-k}$ 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.