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This paper derives an interior $C^2$ estimate for solutions to a specific 2-sum Hessian equation involving $\sigma_2$ and $\sigma_1$ terms, under conditions $\alpha \ge 0$ and $D^2\alpha = \beta I$. The core technique involves transforming the original equation into a higher-dimensional quadratic Hessian equation by introducing an auxiliary variable and a quartic correction, which then allows the application of existing interior estimates. The analysis also covers the gradient-independent case and provides a counterexample demonstrating that an $L^\infty$ bound on $\alpha$ is insufficient for such estimates.
A novel transformation unlocks interior $C^2$ estimates for a complex 2-sum Hessian equation, previously intractable.
This paper establishes an interior $C^2$ estimate for solutions of \[ \sigma_2(D^2u)+\alpha(x)\sigma_1(D^2u)=f(x,u,Du)>0, \] on the positive branch $\Delta u>0$, under the assumptions $\alpha\ge0$ 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 $D^2u$ 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.