Interior Hessian estimates for Hessian quotient equations in dimension three

TL;DR

This paper establishes interior Hessian estimates for 2-convex solutions to Hessian quotient equations in three dimensions, introducing new methods and extending results to higher dimensions with semi-convexity conditions.

Contribution

It provides novel interior Hessian estimates for 2-convex solutions in 3D and extends these estimates to higher dimensions using a new doubling inequality method.

Findings

Interior Hessian estimates for 2-convex solutions in 3D

Doubling inequality established for dimensions 3 and 4

Hessian estimates extended to higher dimensions with semi-convexity

Abstract

In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{\sigma_2}{\sigma_1} (D^2 u) = \psi (x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions ($n\geq 5$) the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation $\sigma_2 (D^2 u) = \psi (x, u, \nabla u)$.