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

Abstract

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.