Interior Hessian estimates for Hessian quotient equations in dimension three

Abstract

In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{\ddot{I}{ƒ}_2}{\ddot{I}{ƒ}_1}(D^{2}u)=\ddot{I}ˆ(x,u)$ in dimension three. In higher dimensions ($n\ge 4$), 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 ($n\ge 5$) 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 $\ddot{I}{ƒ}_2(D^{2}u)=\ddot{I}ˆ(x,u,\nabla u)$.