两类蒙日-安培型方程的研究

In this Programme, we study two kinds of elliptic equations arised from gemetry problems: one is the Monge-Ampère equation which is singular at the boundary of the domain, the other is Monge-Ampère type equation on the unit sphere. For the first one, we mainly study the optimal regularity of the solution and the geometry body denoted by the solution, emphasizing the analysis of the sigularity for the the solution and the boundary of the geometry body and making clear the constructure of the singularity. For the latter, we mainly study the existence and uniqueness of the smooth solutions for the different classes of the given curvature functions. Because of the fully nonlinearity and singularity for the first and the no a priori estimates and the critical case for the latter, this programme requires a great of original idea and method to overcome the difficulities. We hope this programme not only solves the problems related the two kinds of equations, but also provided new methods and techniques for similar elliptic and parobolic problems.

本项目拟研究源于几何问题的一类在区域边界上奇异的Monge-Ampère方程和一类球面上的Monge-Ampère型方程。对于前者，主要研究其解以及该解所表示的几何体的最佳正则性，重点是对解和相应几何体的边界点做奇性分析，也就是将解在区域边界点附近展开，弄清楚展开式中的奇异项的构造。对于后者， 主要研究对于给定的不同类型的曲率函数， 研究该方程光滑解的存在性和唯一性。 由于前者的完全非线性和奇异性， 而后者即没有先验估计又是处在临界指数的情况， 所以本项目在分析上的难度是很大的， 需要方法和技巧上的很多创新。通过本项目的研究， 我们期望不但能解决与这两类方程相关的问题，而且能将其解决过程所获得的新方法和新技巧应用于其它奇异的、既没有先验估计又是临界指数情况的非线性椭圆或抛物问题。