An Integral Approach to Prescribing Scalar Curvature Equations

Abstract

We develop an integral approach to obtain interior a priori $C^{1,1}$ estimates for convex solutions of prescribing scalar curvature equations ${\sigma }_2(\kappa )=f(x)$ as well as the Hessian equations ${\sigma }_2(D^{2}u)=f(x)$. This new approach can deal with the case when $f$ is of weaker regularity. As a result, we prove that the $C^{1,1}$ modules of the solutions depend only on the Lipschitz modules of $f(x)$, instead of the $\Vert f{\Vert }_{C^k}$ for some $k\ge 2$ in all the papers we have known up to now.