Interior $C^2$ Estimates for 2-Sum Hessian Equations

Abstract

This paper establishes an interior $C^2$ estimate for solutions of \[ σ_2(D^2u)+α(x)σ_1(D^2u)=f(x,u,Du)>0, \] on the positive branch $\Delta u>0$, under the assumptions $\alpha \ge 0$ and $D^2\alpha =\beta I$. Using one extra variable and a quartic correction, the equation is reduced to a quadratic Hessian equation in one higher dimension. The terms involving $D^{2}u$ cancel, so the interior estimate of Li--Wu \cite{LW} applies. The gradient-independent case is also treated. Finally, an example shows that an $L^{\infty }$ bound on $\alpha$ alone is not enough.