Estimates for k-Hessian operator and some applications

The k-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations F _k u] = 0, where F _k u] is the elementary symmetric function of order k, 1 ? ? 6 n, of the eigenvalues of the Hessian matrix D ² u. For example, F ₁u] is the Laplacian Δu and F _n u] is the real Monge-Ampère operator detD ² u, while 1-convex functions and n-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative k-convex functions, and give several estimates for the mixed k-Hessian operator. Applications of these estimates to the k-Green functions are also established.