C2 estimates for k-Hessian equations and a rigidity theorem

Abstract

We derive a concavity inequality for k-Hessian operators under the semiconvexity condition. As an application, we establish interior estimates for semiconvex solutions to the k-Hessian equations with vanishing Dirichlet boundary conditions and obtain a Liouville-type result. This result confirms Chang-Yuan's conjecture [4] under the super quadratic growth condition.