The regularity for the Monge-Ampere equation and the k-Hessian equation

In this thesis, we study some important nonlinear partial differential equations, including the Monge-Ampere equation, the k-Hessian equation and the k-curvature equation. There are four problems studied in this thesis. Chapter 2 concerns the existence and uniqueness of Alexandrov’s solutions for the Dirichlet problem of the Monge-Ampere equation by the continuity method. Chapter 3 contains a new proof for the interior C2,α regularity of the Monge- Ampere equation under the assumption sup Ω |D2u(x)| ≤ Λ by using the Green function. Chapter 4 presents the interior C1,α regularity for the k-Hessian equation and the k-curvature equation with the boundary condition u = 0 on ∂Ω. Finally, in chapter 5, we present the global C1,α regularity for the k-Hessian equation and the k-curvature equation with the boundary condition u = φ on ∂Ω.