Boundary Regularity for k-Hessian Equations

Abstract

In this paper we focus on the boundary regularity for a class of k-Hessian equations which can be degenerate and (or) singular on the boundary of the domain. Motivated by the case of Monge–Ampère equations, we first construct sub-solutions, then apply the characteristic of the global Hölder continuity for convex functions, and finally use the maximum principle to obtain the boundary Hölder continuity for the solutions of the k-Hessian equations. However, finding such sub-solutions is very difficult due to the complexity of the k-Hessian operator. In particular, we employ the symmetric mean to overcome the difficulties.