Pogorelov Type C~2 Estimates For Sum Hessian Equations And A Rigidity Theorem

In this paper,we mainly study Pogorelov type C2 estimates of solutions for the Dirichlet problem of Sum Hessian equations in the following forms:Here,u is an unknown function defined on Ω.Denote Du and D2u to be the gradient and the Hessian of u.α is a positive constant,we also require f>0 and smooth enough with respect to every variables.σk（D2u）=σk（λ（D2u））denotes the k-th elementary symmetric function of the eigenvalues of the Hessian matrix D2u.Namely,for λ=（λ1,…,λn）∈Rn,First,we establish Pogorelov type C2 estimates of Sum Hessian equations（0.1）for admissible solutions under some conditions;Second,we establish Pogorelov type C2 estimates of Sum Hessian equations（0.1）for k-convex solutions;Finally,we apply such estimates to obtain a rigidity theorem for k-convex solutions of Sum Hessian equations in Euclidean space.