Convexity and asymptotic estimates for large solutions of Hessian equations

Abstract

We consider the smallest viscosity solution of the Hessian equation $S_k(D^2u)=f(u)$ in $\omega$, that becomes infinite at the boundary of $\omega\subset\mathbb R^n$; here $S_k(D^2u)$ denotes the $k$-th elementary symmetric function of the eigenvalues of $D^2u$, for $k\in\{1,\dots, n\}$. We prove that if $\omega$ 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 of $\omega$.