Strictly 3-convex estimates for sigma-3 quotient equations

Establish interior \(C^2\) estimates for strictly 3-convex solutions of the Hessian quotient equations \(\sigma_3(D^2u)/\sigma_l(D^2u)=1\) for \(l=1,2\) in all dimensions.

Background

A cited transformation reduces semi-convex solutions of the equations $\sigma_3/\sigma_l=1$, with $l=1,2$, to semi-convex solutions of suitable sum-Hessian equations. This yields interior $C^2$ estimates for the semi-convex class in all dimensions.

The paper notes that the corresponding estimates under the weaker and different strictly 3-convex assumption had not been established. Thus the open problem is to extend the semi-convex estimates to strictly 3-convex solutions.