Interior Hessian estimates for sum Hessian quotient equation

TL;DR

This paper develops interior second derivative estimates for sum Hessian quotient equations, including Pogorelov type estimates, advancing the understanding of regularity for these nonlinear PDEs.

Contribution

It introduces new interior $C^2$ and Pogorelov type estimates for sum Hessian quotient equations, including special cases when $k=n$, filling gaps in regularity theory.

Findings

Established interior $C^2$ estimates for $0 extless l extless k extless n$

Derived Pogorelov type estimates for the same class of equations

Obtained weaker Pogorelov estimates when $k=n$ and $0 extless l extless n-1$

Abstract

This paper is devoted to the interior $C^2$ estimates for a class of sum Hessian quotient equations. For $0\leq l<k<n$, we establish the interior estimates and the Pogorelov type estimates. In the case $k=n$, we obtain a weaker Pogorelov type estimate for $0\leq l<n-1$.