The Cauchy-Neumann Problem For Parabolic Monge-Ampère Type Equations

Monge-Ampère type equations are very important members of fully nonlinear sec-ond order partial differential equations.They were first presented by Monge^[23]and Ampère^[1].Later,Bernstein,Pogorelov,Nirenberg and others made a deeper research.Monge-Ampère type equations originated from optimal transportation problems,and they are applied to differential geometry widely,for example:Weyl problem^[22]and Minkowski problem^[32].And they are applied to affine geometry,conformal geometry and geometrical optics,etc.Therefore we shall have a deeper understanding on non-linear partial differential equations with further investigation on Monge-Ampère type equations.In this paper,we study Cauchy-Neumann problem of Monge-Ampère type equa-tions.By constructing auxiliary function,using the maximum principle,we have ob-tained maximum modulus estimates,gradient estimates and second order derivative estimates.Finally,we get the existence of solutions by continuity method.This dissertation includes three section.In section 1,we present the background and main results of Monge-Ampère type equations.In section 2,we introduce the strong maximum principle,weak maximum principle,a priori estimates and some es-sential marks and definitions which are related to Monge-Ampère type equations.In section 3,we proof maximum modulus estimates,gradient estimates and second or-der derivative estimates for solutions then we illustrate the existence,uniqueness and regularity for solutions by basic theory for parabolic equations and continuity method.