On The Dirichlet Problem For Some Monge-Ampere Type Equations

In this thesis, we study an important kind of fully nonlinear partial differential equations: Monge-Ampere type equations. These equations arise from optimal transportation, geometric optics, conformal geometry and etc. The Monge-Ampere type equations under consideration in this thesis have the following form: det[D2u-A(x, u, Du)]=B(x, u, Du), where A is a matrix function, the righthand side scalar function B>0. When the matrix function A(?)0, the equations become the standard Monge-Ampere equations.There are two main problems studied in this thesis. One problem is to study the classi-cal solvability of the Dirichlet problem for the Monge-Ampere type equations via continuity method. Under the assumptions that A is regular (i.e. A3w) and the existence of a smooth subsolution, we obtain a priori derivative estimates up to second order for the solutions by the standard barrier arguments. Global barrier functions are constructed by the smooth subsolution under minimal hypotheses. Since the construction process is very technical, it is a difficulty in this thesis. Also, under the same assumptions, similar existence results are extended to the Dirichlet problem for a class of augmented Hessian equations, where we make full use of differ-ent properties for the Hessian operator. The other problem is to establish an equivalence result for Aleksandrov generalized solutions and viscosity solutions to a kind of optimal transporta-tion equations. To prove this, the above classical existence result of the Dirichlet problem for Monge-Ampere type equations is used.In this thesis, we obtain the existence of the solution by the existence of a subsolution. This method to deal with the Dirichlet problem is called the subsolution method. By such method, we can construct barrier functions directly by using the smooth subsolution. Also, the barrier construction do not rely on the geometric properties of the domain. Therefore, we can study the Dirichlet problems on smooth bounded domains without other geometric restrictions.We only assume the matrix function A is regular and the existence of a smooth subsolution in obtaining the second derivative bounds. Since the regular condition for the matrix function A is necessary for C1regularity of the solution and the existence of a smooth subsolution is necessary for the subsolution method, the second derivative estimates established in this thesis are sharp in some sense.The Monge-Ampere type equations arising from optimal transportation, geometric optics and conformal geometry has different special forms. These equations all have the general form studied in this thesis. Therefore, the existence theorems for the Dirichlet problems of Monge-Ampere type equations studied here can be applied to many equations arising from different problems.