Convexity and asymptotic estimates for large solutions of Hessian equations

We consider the smallest viscosity solution of the Hessian equation $S_k(D^2u) = f(u)$ in a k-convex domain­, that becomes infinite at the boundary of the domain; here $S_k(D^2u)$ denotes the k-th elementary symmetric function of the eigenvalues of the Hessian matrix of the function $u$, for k ∈ {1, . . . , n}. We prove that if the domain­ is strictly convex and $f$ satisfies suitable assumptions, then the smallest solution is convex. We also establish asymptotic estimates for the behaviour of such a solution near the boundary.