Doctor of Philosophy
School of Mathematics and Applied Statistics
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.
Wu, Yating, The regularity for the Monge-Ampere equation and the k-Hessian equation, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2022. https://ro.uow.edu.au/theses1/1450
FoR codes (2008)
0101 PURE MATHEMATICS
Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.