The mean curvature estimate for the level sets of solutions of the Monge-Ampère equation on Riemannian manifold

Abstract

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.

Convexity Estimates for the Solution of the Elliptical Partial Differential Equation det D 2 u = eu

“… related to the strictly convex solution u is constructed to make it reach the maximum value at the boundary. [3] Let ( )…”