An Integral Approach to Prescribing Scalar Curvature Equations

TL;DR

This paper introduces an integral method to derive interior $C^{1,1}$ estimates for convex solutions of scalar curvature and Hessian equations, accommodating weaker regularity of the prescribed function $f(x)$.

Contribution

It presents a novel integral approach that allows for $C^{1,1}$ estimates depending only on the Lipschitz norm of $f$, improving upon previous regularity requirements.

Findings

Established interior $C^{1,1}$ estimates for convex solutions.

Demonstrated dependence of solution regularity solely on Lipschitz norm of $f$.

Extended applicability to cases with weaker regularity of $f$.

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 $\sigma_2(D^2u) = f(x)$. 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 $\|f\|_{C^k}$ for some $k\geq 2$ in all the papers we have known up to now.