有关四阶Monge-Ampere型方程若干问题的研究

项目名称： 有关四阶Monge-Ampere型方程若干问题的研究

项目编号： No.11301034

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 鞠红杰

作者单位： 北京邮电大学

项目金额： 22万元

中文摘要： 仿射极大曲面方程和Abreu方程均为四阶Monge-Ampere方程，是微分几何中两个非常热门的研究课题。数学大师陈省身和李安民院士分别关于上述两个方程提出猜想：整凸解是二次函数。由于方程的高阶性和非线性性给这类方程的研究带来很大困难，许多问题都没有解决，例如上述猜想的高维情形(n>2)目前仍是公开的难题。本项目研究较仿射极大曲面方程和Abreu方程更广泛的一类四阶Monge-Ampere方程。我们拟首先考虑弱解的正则性问题，关键是得到二阶导数估计以及二阶导数的Holder估计。然后结合正则性结果研究方程有界区域边值问题可解性，外Dirichlet问题，以及解在无穷远处的渐近性问题。最后拟通过对渐近性质的研究，并结合Trudinger和汪徐家等专家的分析技巧探讨整解的Bernstein性质，从而推进高维情形陈省身猜想和李安民猜想的最终解决。

中文关键词： Monge-Ampere 方程；正则性；存在性；无界区域；

英文摘要： Affine maximal surface equation and Abreu equation are two fourth order Monge-Ampere equations,which are hot research topics in diferential geometry.For affine maximal surface equation and Abreu equation, Chern S.S. and Li A.M. conjectured that the entire convex solution must be quadratic function, respectively. The higher order and nonlinearity of the class of equations cause great difficulties for the research, and there are many issues unresolved,for example,the above two conjectures in high dimension are open problems.This project is intended to study a class of fourth order Monge-Ampere type equations,including affine maximal surface equation and Abreu equation.Firstly,we consider the regularity of weak solutions to the equation.The key is to obtain second derivative estimate and Holder estimate of the second derivative.Then we study the solvability of boundary value problem,exteiror Dirichlet problem and the asymptotic behavior of the solution at infinity.Furthermore, we will use the analytical technique of Trudinger, Wang X.J.and other experts, and the asymptotic behavior to investigate the Bernstein property of entire solution,thereby improving the final resolution of Chern's conjecture and Li's conjecture.

英文关键词： Monge-Ampere equation；regularity；existence；unbounded domain；