The Dirichlet Problem For A Class Of Singular Complex Monge-Amp(?)re Equations

In this thesis,we consider the Dirichlet problem of the Monge-Amp(?)re equa-tions:(?) where 0<F∈C∞((B1),0<α<n and φ∈C∞((?)B1).Based on the work of Guedj-Kolodziej-Zeriahi,we have known that there exists a unique Holder continuous solution to this equation.We consider one family of approximate equations:(?)We hope to get some regularity results of the solution to the original equation,i.e.the solution is smooth except the origin,by establishing the C1-estimate and C2-estimate of((u(ε)),the solutions to approximate equations for weighted versions.Further,we will give a little improvement for the Holder exponent of the solution to the original equation.In Chapter 2,we recall some basic definitions and theorems;In Chapter 3,we get the uniform C0-estimate of the solutions to approximate equations;In Chapter 4,we get the gradient estimate with weight of the solutions to approximate equations;In Chapter 5,we get the C2-estimate with weight of the solutions to approx-imate equations;In Chapter 6,we get one regularity of the solution to the original equation;In Chapter 7,we give one improvement for the Holder exponent of the solu-tion to the original equation;In Chapter 8,we list some questions we are interested in;Last,we provide one C2,α-estimate method in appendix.