两类Monge-Ampere方程问题的研究

项目名称： 两类Monge-Ampere方程问题的研究

项目编号： No.11271118

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 孙明保

作者单位： 湖南理工学院

项目金额： 68万元

中文摘要： 本项目研究两类Monge-Ampere方程，其中一类是仿射几何中极大曲面方程和预定仿射平均曲率方程，它们是全非线性四阶Monge-Ampere型方程，本项目将研究该方程在无界区域中解的存在性、正则性和在无穷无处的渐近性质，包括陈省身关于仿射极大曲面著名猜想的高维情况；另一类是与满足Hormander条件向量场相关的Monge-Ampere方程，本项目将研究该方程解的正则性，其中包括解的C^{1,α}正则性、解的C^{2,α}估计和W^{2,p}估计问题。

中文关键词： Monge-Ampere方程；平均曲率；向量场；正则性；Minkowski问题

英文摘要： In this project，we intend to study two kinds of Monge-Ampere equations. One is the maximal surface equation and the equation of prescribed affine mean curvature in affine geometry, which are fully nonlinear fourth order equations of Monge-Ampere type. We will study the existence, regularity and asymptotic properties of the solutions at infinity for the unbounded domain, which includes the famous Chern conjecture for the affine maximal hypersurfaces in the high dimensional case. The other is Monge-Ampere equations associated with vector fields satisfying Hormander's condition. This project will study the regularity of the solutions of the equations, which includes the C^{1,α} regularity, C^{2,α} estimates and W^{2,p} estimates problems.

英文关键词： Monge-Ampere equations；mean curvature；vector fields；regularity；Minkowski problem