The C^1,1 Regularity For Solutions Of Some Degenerate Monge-Ampère (Type)Equations

The Monge-Ampere type equation is a class of very important nonlinear second order elliptic partial differential equations,which is closely related to several problems from analysis and geometry,such as the Minkowski problem,Weyl transportation,etc.In this thesis,we shall study the Monge-Ampère type equation det[D2u-A(·,u,Du)]=B(·,u,Du),in ?,(2)where ? is a bounded domain,Du and D2u denote the gradient vector and Hessian matrix the unknown function u:??R respectively,A:?×R×Rn?Rn×n is a symmetric n×n matrix valued function,B:?×R×Rn?R+?{0} is a nonnegative scalar function.We say Monge-Ampere type equation(2)is degenerate if the right hand function B of the equation(2)is nonnegative.In this thesis,we study some degenerate Monge-Ampere(type)equations.The main results achieved include:interior C1,1 regularity for solutions of the degenerate Monge-Ampere(type)equations,and existence and uniqueness of C1,1 solutions for the degenerate Monge-Ampere(type)equations with the Neumann boundary value problem.Since the equation is degenerate,the first and second order derivatives of log B might be+? when B?0+,which is hard for us to establish the second order derivative estimates for solutions of the degenerate Monge-Ampere type equation(2).To deal with the degeneracy,we need to build the relationship between B-1/n-1 and(?),where ?uij?is the inverse matrix of {uij},uij=uij-Aij.For the interior C1,1 regularity of solutions,we can establish a more general Pogorelov estimate if it is Monge-Ampere equation.If it is the Monge-Ampere type equation,we construct a suitable barrier function by the structure of the Monge-Ampere type equation.Using the fact that the maximal eigenvalue of the matrix {uij-Aij}n×n has lower bound at the maximal point of the barrier function,and building the relationship between B-1/n-1 and(?)by the equation at the maximal point,then we can obtain the Pogorelov type estimates for solutions of the Monge-Ampere type equations(2).Furthermore,we can establish the interior C1,1 regularity for solutions of degenerate Monge-Ampere(type)equations by the Pogorelov(type)estimates.For the global C1,1 regularity for solutions of the degenerate Monge-Ampère equations with the Neumann boundary value condition,it only needs to deal with the degeneracy in the process of reducing the global estimates to the boundary and double normal estimates.Therefore,we should build the relationship between B-1/n-1 and(?)to deal with the degenerate term.Especially,for double normal estimates on the boundary,we construct a barrier function which contains D?u,and split the region into two parts(?)(?)and(?).In(?)?we can build the relationship between B-1/n-1 and(?)by the equation;in(?),we can obtain the contradiction between the property of the barrier function at its maximal point and(?).Therefore,we get the double normal estimates on the boundary.For the global C1,1 regularity of solutions for the degenerate Monge-Ampere type equations with the Neumann boundary value condition,we need to adjust the barrier functions by the property of the Monge-Ampere type equations in order to obtain the global second order derivative estimates.