一类Hessian方程解的凸性研究

解的几何性质是偏微分方程理论中的基本问题，而凸性作为一个重要的几何特征，长期以来一直是偏微分方程研究中的重要主题。本项目的主要研究对象是Hessian方程解的凸性。利用经典的极大值原理给出水平集的凸性刻画，曲率估计，解本身的凸性估计，对于不同Hessian方程寻找其特定的曲率或凸性估计函数，建立其满足的微分不等式， 描述给定辅助函数的最大值或最小值在边界达到。通过边界给定的已知条件和获得的辅助函数的结果，得到所研究偏微分方程的解的具体的几何信息,特别是其凸性信息。

The geometric properties of solutions are the fundamental problems in the theory of partial differential equations. As one of the most important geometric properties, convexity has been an issue in the study of elliptic partial differential equations for a long time. In this project, the main research problem is the convexity of solutions for a class of Hessian equaitons. We hope to obtain the convexity descriptions and curvature estimates of the level sets of the solutions and the convexity estimates of the solutions themselves. For the different Hessian equations, we will look for the particular auxiliary functions of curvature and convexity estimates, build up the differential inequalities of the auxiliary functions, and describe that the functions attain their maximum or minimum on the boundary. We plan to get the specifically geometric information, especially convexity, of the solutions to partial differential equations to be researched through the given boundary conditions and the results we will obtain for the auxiliary functions.