Regularity for convex viscosity solutions of ${\sigma }_3$ equation

Abstract

We prove interior $C^2$ regularity for convex viscosity solutions of the $3$-Hessian equation ${\sigma }_3(D^{2}u)=f(x)$ with $f\in C^{0,1},\inf f>0$, under a strict $3$-convexity condition on $u$.