Global $W^{2,p}$ and $C^{2,\alpha }$ regularity for the sigma-$2$ equation

In this paper, we prove that continuous $2$-convex solutions of the ${\sigma }_2$ equation with positive continuous density on bounded strictly mean-convex $C^3$ domains with $C^3$ boundary values belong to $W^{2,p}$ for every $1<p<\mathrm{\infty }$ in every dimension. If, in addition, the density is $C^{\alpha }$, $0<\alpha <1$, the solutions belong to $C^{2,\alpha }$. Moreover, two counterexamples show that neither the $C^3$ assumption on the domain nor that on the boundary values can in general be weakened to $C^{2,1}$.