Interior $C^2$ estimate for Hessian quotient equation in general dimension

TL;DR

This paper investigates the interior second derivative regularity for the Hessian quotient equation, establishing estimates for certain cases and demonstrating failure in others through singular solutions.

Contribution

It provides a complete characterization of interior $C^2$ estimates for the Hessian quotient equation across different dimensions and parameters.

Findings

Established interior $C^2$ estimates for $k=n-1,n-2$

Proved failure of interior $C^2$ estimates for $k \,\leq\, n-3$

Constructed singular solutions demonstrating the failure cases

Abstract

In this paper, we study the interior $C^2$ regularity problem for the Hessian quotient equation $\left(\frac{\sigma_n}{\sigma_k}\right)(D^2u)=f$. We give a complete answer to this longstanding problem: for $k=n-1,n-2$, we establish an interior $C^2$ estimate; for $k\leq n-3$, we show that interior $C^2$ estimate fails by finding a singular solution.