Regularity for convex viscosity solutions of $σ_2$ Equation

Abstract

We prove interior $C^{2}$ regularity result for convex viscosity solutions of the quadratic Hessian equation $σ_2(D^2u) = f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^α$ for some $α\in (0,1)$, the corresponding interior regularity remains open.