Interior \(C^2\) Estimate for Hessian Quotient Equation in General Dimension

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.