Convexity and asymptotic estimates for large solutions of Hessian equations

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