Notes on generalized special Lagrangian equation

We obtain a priori \(C^{1,1}\) $C^{1,1}$ estimates for some Hessian quotient equations with positive Lipschitz right hand sides, through studying a twisted special Lagrangian equation. The results imply the interior \(C^{2,\alpha }\) $C^{2,\alpha }$ regularity for \(C^0\) $C^0$ viscosity solutions to \(\sigma _2=f^2(x)\) ${\sigma }_2=f^2(x)$ in dimension 3, with positive Lipschitz f(x).