Curvature estimation of the level set of solutions of the elliptic Hessian equation

Abstract Hessian equations are an important class of nonlinear partial differential equations. The geometric convexity estimation of PDE solutions and the estimation of the level set of solutions are important research topics in elliptical partial differential equation theory. Curvature is widely used to describe the degree of curvature of curves, which is a very important basis in the study of differential geometry. We will study the completely nonlinear elliptical Hessian equation σ n ( D 2 u ) = e u in n-dimensional Euclidean space. Then, the auxiliary functions, which are directly related to the level set of the solutions, are established. Under the condition of 0 boundary Dirichlet, with the help of the Maximum principle, we will make an upper bound estimate of the curvature for the level set of the solutions. The main contents of this paper are mean curvature and Gauss curvature.