The k-Hessian Equation

The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k >= 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a gradient flow method we prove a Sobolev type inequality for k-admissible functions vanishing on the boundary, and study the corresponding variational problems. We also extend the definition of k-admissibility to non-smooth functions and prove a weak continuity of the k-Hessian operator. The weak continuity enables us to deduce a Wolff potential estimate. As an application we prove the Holder continuity of weak solutions to the k-Hessian equation. These results are mainly from the papers [CNS2, W2, CW1, TW2, Ld] in the references of the paper.