Interior $C^{2,\alpha }$ Regularity for Convex Solutions of the 2-Hessian Equation

Abstract

We consider convex admissible viscosity solutions of ${\sigma }_2(D^{2}u)=f>0$ in an open subset of ${\mathbb{R}}^n$, where $n\ge 2$, $0<\alpha <1$, and $f\in C_{\mathrm{loc}}^{0,\alpha }$. We prove that every such solution belongs to $C_{\mathrm{loc}}^{2,\alpha }$. For solutions in $B_2$, we also prove a uniform $C^{2,\alpha }(B_{1/4})$ estimate under an $L^{\infty }$ bound for $u$, a positive lower bound for $f$, and a $C^{0,\alpha }$ bound for $f$.