A Priori Estimates for Monge-Ampere Equation and Applications

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

A unified flow approach to smooth, even LpL_p-Minkowski problems

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in $\mathbb{R}^{n+1}$ for $p>-n-1.$Comment: 21 pages. Comments are welcom

On the uniqueness of LpL_p-Minkowski problems: the constant pp-curvature case in R3\mathbb{R}^3

We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is a constant. In the case of $n=2$, either when $p\in[-1,0]$ or when $p\in(0,1)$ in addition to a pinching condition, we show that $\mathbb{K}$ must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the $L_p$-Minkowski problem in $\mathbb{R}^3$. Moreover, we give an explicit pinching constant depending only on $p$ when $p\in(0,1)$.Comment: references update

Orlicz-Minkowski flows

We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding $L_p$ version is the even $L_p$-Minkowski problem for $p>-n-1$. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the $L_p$ versions are the even $L_p$-Minkowski problem for $p>0$ and the $L_p$-Minkowski problem for $p>1$. In the final section, we use a curvature flow with no global term to solve a class of $L_p$-Christoffel-Minkowski type problems.Comment: 30 page