A Differential Inequality for the Solution of Monge Ampère Equation on Riemannian Manifold

Abstract

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.

Curvature estimation of the level sets of solutions of a class of elliptic partial differential equation

“….It is elliptic when the Hessian matrix is positive definite. In reference [5] [6], the strict convex solutions of the equation are studied in Riemann manifolds, and the mean curvature estimates of the level sets of the solutions are obtained.…”