The Neumann Boundary Value Problems For Complex Monge-Ampère Equations

Complex Monge-Ampere equations originate from pluripotential theory, the fa-mous conjecture of Calabi in differential geometry and physics. The problems involve important fields of multicomplex function theory, differential geometry and fully non-linear partial differential equations.There are a lot of papers for the Dirichlet boundary value problems. While, for the Neumann boundary value problems, only in 1994, Li Songying proved the existence, uniqueness and regularity of the solutions in [13]. In this paper,we give the new proof for the gradient estimates of the solutions subject to the complex Monge-Ampere equa-tions with the Neumann boundary value problems. Li Songying proved the gradient estimates by the method of auxiliary function reduced wholly to the boundary in [13]. In this paper, firstly, we suppose the existence of the gradient estimates; then, we rewrite the proof of the second-order derivatives estimates by the train of the thought in [13]; finally, we use the method of interpolation inequality to obtain the global estimates of the gradient.This paper is divided into four parts. Firstly, we introduce the origination, devel-opement and meaning of the problems. Continiously, we set the stage introducing basic symbols, maximum principles, comparison principles and the method of continuity. Then, we introduce the conclusions of the real Monge-Ampere equations with oblique boundary value problems.In addition,we show the proof of the gradient estimates of the solutions for the complex Monge-Ampere equations with Neumann boundary val-ue problems in [14]. Finally, we give the new proof of the gradient estimates...