Second order estimates for convex solutions of degenerate $k$-Hessian equations

Abstract

The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)}\in C^{1,1}$. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain.