For the fully nonlinear elliptic Monge-Ampere equation det D 2 u = 1 with homogeneous Dirichlet boundary value condition, in this paper, a function related to the curvature of the level set of the solution was established, then the differential inequality of the strictly convex solutions of the equation on four-dimensional Riemannian manifold was got. The maximum value of the auxiliary function at the boundary was obtained by using the maximum principle and the mean curvature estimation for the level sets of the solution was given.

In three-dimensional Riemannian manifolds with constant curvature, the elliptic Monge amp è re equation satisfying the homogeneous Dirichlet boundary value condition is studied. Under certain conditions, an estimate related to the solution of the equation is made, and a detailed proof of differential inequality is given.

The curvature of the level set of elliptic partial differential equation solutions is always an important content in the study of convexity. Curvature is an important invariant of surface, which characterizes the degree of curve bending, is the important basis of differential geometry. Curvature is widely used in machining. In this paper, we study the completely nonlinear elliptic Monge-Ampère equation 2 det u D u e = with 0 boundary value Dirichlet condition in four-dimensional Euclidean space. It is proved that the auxiliary function obtains the maximum value at the boundary, and then the mean curvature and Gauss curvature of the level sets of the solutions of the equation are estimated quantitatively.

The Monge-Ampère equation det D 2 u = eu is completely nonlinear and elliptic, the convexity estimates for the solution of the elliptical partial differential equation is very important. We establish a differential inequality by constructing an auxiliary function and give two differential estimates for the solution of the equation det D 2 u = eu with 0 boundary value condition.