An Integral Approach to Prescribing Scalar Curvature Equations

We develop an integral approach to obtain interior a priori C1,1 estimates for convex solutions of prescribing scalar curvature equations σ₂(κ) = f(x) as well as the Hessian equations σ₂(D²u) = f(x). This new approach can deal with the case when f is of weaker regularity. As a result, we prove that the C1,1 modules of the solutions depend only on the Lipschitz modules of $f(x)$, instead of the \|f\|Cᵏ for some k≥ 2 in all the papers we have known up to now.