Some Boundary Value Problems For Complex Monge-Ampère Equations

Complex Monge-Ampere equations are very important topics that deal with complex analysis, differential geometry, partial differential equations and physics etc. Pluripoten-tial theory and the Calabi conjecture in differential geometry are the sources of complex Monge-Ampere equations.The classical form of complex Monge-Ampere equations is a fully non-linear elliptic equation:where the solution u is required to be a plurisubharmonic function in some open subsetΩof C~n,f>0.During the last twenty years complex Monge-Ampere equations have been a subject of intensive studies. The main aim of this thesis is to study some boundary value problems for complex Monge-Ampere equations, which contain the semilinear oblique boundary problem, infinite boundary value problem and Dirichlet boundary problem in some special domain that is not strictly pseudoconvex.In Chapter 1 we recall the backgrounds and give a survey of the study on complex Monge-Ampere equations. Then we give an outline of this thesis including our main problems, results and arguments.In Chapter 2, we introduce some preliminaries, notations and main results about Dirichlet and Neumann problems firstly. Then we prove the existence and uniqueness of classical plurisubharmonic solutions for complex Monge-Ampere equations subject to the semilinear oblique boundary problem in the certain strictly pseudoconvex domain.In Chapter 3, at first, we introduce the work by Cheng and Yau that connect Calabi conjecture with complex Morjge-Ampere equations. Then we obtain the existence and nonexistence of the solutions for complex Monge-Ampere equations and Hessian equationswith infinite Dirichlet boundary value in the certain bounded domain in C~n.In Chapter 4, we introduce the basic properties for pluripotential theory and results about the existence of weak solutions. Then we prove the existence of weak solution in some special domain which is not strictly pseudoconvex.We mainly use the barrier functions, the interpolation inequality, a prior estimate and continuity method in this thesis.