Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides

In this paper, let $u\in C^4(B_{10})$ with $D^{2}u\ge -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset {\mathbb{R}}^{n+1}$ satisfying $${\sigma }_2(\kappa [u])=f(x)\mathrm{.}$$ We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality for $b=\log (H+J_0)$, a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted $L^1$ estimate, which is completed using the Jacobi energy inequality and integration by parts.