A Priori Estimates for Monge-Ampere Equation and Applications

Abstract

In this thesis we study different applications of the Monge-Ampere type equations. Chapter 1 is an introduction. In Chapter 2, we study the convergence rate of discrete Monge-Ampere type equation. In Chapter 3 we study the Lp-dual Minkowski problem. In Chapter 4 we study the asymptotic affine hyperspheres.

The numerical solution to Monge-Ampere equation, in particular the Dirichlet problem has drawn much attentions in last 20 years. Different algorithms have been designed to simulate numerical solutions. We approximate the solution u by a sequence of convex polyhedra, which are generalised solutions to the Monge-Ampere type equation in the sense of Aleksandrov, and the associated Monge-Ampere measure are supported on a properly chosen grid in the domain. We derive the convergence rate estimates for the cases when f is smooth, Holder continuous and merely continuous in Chapter 2.

Lp dual Minkowski problem is introduced by Lutwak-Yang-Zhang recently, which amounts to solving a class of Monge-Ampere type equations on the sphere. Our main purpose in Chapter 3 is to solve the Lp dual Minkowski problem in the case for all p > 0 studying of related parabolic flows. Also under these flows we obtain some uniqueness, smoothness and positivity results for the problem. We generalise a Blaschke-Santol`o type inequality, and applied the inequality in a variational method to obtain some existence and non-uniqueness results for the problem in the symmetric case.

In Chapter 4, we study a singular Monge-Ampere type equation related to affine hyperspheres. We show the existence of solutions via regularization method, followed by the existence of asymptotic affine hyperspheres. Also, we study the regularity of this Monge-Ampere type equation and obtain the optimal graph regularity.

The results are contained in the published papers.