众所周知，先验估计是研究椭圆偏微分方程的存在性，唯一性和正则性等基本问题的基础，也是完全非线性椭圆方程研究中的极为重要的问题。二阶导数估计又是先验估计中至关重要的一步。k-Hessian 方程是完全非线性椭圆方程的主要研究对象之一。本项目研究带有一般右端项的 k-Hessian 方程的全局二阶导数估计。申请人已经和加拿大 McGill大学的管鹏飞教授，吉林大学任长宇副教授合作，建立了有一般右端项 k-Hessain 方程的凸解的全局二阶导数估计，也建立了有一般右端项的 2-Hessain方程的 2-凸解的全局二阶导数估计。申请人和任长宇副教授一起合作，建立了有一般右端项的 n-1-Hessain方程的 n-1-凸解的全局二阶导数估计。本申请项目期望对于一般右端项的 k-Hessain 方程建立 k-凸解的全局二阶导数估计，也期望为我们已经建立的估计寻求分析和几何上的新应用。

It is well known that a prior estimates are the foundation of the existences, uniqueness and regularity of the elliptic partial differential equations. Thus, it is also an extremely important problem for the fully nonlinear elliptic partial differential equations. The global estimates of the second derivatives are the key step for a prior estimates. k-Hessian equations are the one of central topics for the fully nonlinear elliptic partial differential equations. Our project plans to study the global estimates of the second derivatives of the k-Hessian equations with general right hand functions. The applicant joints with professor Pengfei Guan in McGill University, Canada and associated professor Changyu Ren in Jilin University to establish the global estimates of the second derivatives of the k-Hessian equations with general right hand functions for convex solutions. We also have established the global estimates of the second derivatives of the 2-Hessian equations with general right hand functions for 2-convex solutions. The applicant also joints with associated professor Changyu Ren to establish the global estimates of the second derivatives of the n-1-Hessian equations with general right hand functions for n-1-convex solutions. In the present project, our goal is to establish the global estimates of the second derivatives of the k-Hessian equations with general right hand functions for k-convex solutions. We also wish to find more new applications of our established estimates for the theory of the elliptic PDEs or geometrical problems.