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

This paper establishes an interior $C^2$ estimate for solutions of $${\sigma }_2(D^{2}u)+\alpha (x){\sigma }_1(D^{2}u)=f(x,u,Du)>0,$$ on the positive branch $\mathrm{\Delta }u>0$, 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 $L^{\mathrm{\infty }}$ bound on $\alpha$ alone is not enough.