Orlicz-Minkowski问题及相关的Monge-Ampere型方程

In recent years, the so-called "Orlicz-Brunn-Minkowski theory" in convex geometry has been built up gradually and developed rapidly. It is the recent development of the classical Brunn-Minkowski theory, and has attracted great attention from many scholars at home and abroad. The Orlicz-Minkowski problem is a basic primary problem in this new theory, and associated Monge-Ampere type equations themselves have many important applications. Since these equations are fully nonlinear partial differential equations on the hypersphere, they are very difficult and quite challenging. Besides they have been studied for only a few years, so there are a lot of problems to be solved. This project aims to solve some important issues about the Orlicz-Minkowski problem and associated Monge-Ampere type equations. Specifically, we are firstly to find a general sufficient condition for the existence of solutions to these equations. Then we construct related curvature flows and analyze some properties of their solutions, such as long time existence, asymptotic behavior, and singularity and so on. Finally, we study the uniqueness of solutions to these Monge-Ampere type equations, especially in certain special cases.

最近几年，凸几何中所谓的“Orlicz-Brunn-Minkowski 理论”逐步建立并快速发展，它是经典 Brunn-Minkowski 理论的最新发展，已引起国内外众多学者的极大关注。Orlicz-Minkowski 问题是这一新理论中首要的基本问题之一，且与此相关的一类 Monge-Ampere 型方程，本身也具有许多重要的应用。由于这类方程是超球面上的完全非线性偏微分方程，具有相当的难度和挑战性，并且它引起大家的关注也不过几年时间，因此有大量问题亟待解决。本项目拟解决 Orlicz-Minkowski 问题及相关的 Monge-Ampere 型方程中的某些关键问题。具体地说，我们拟首先寻找这类方程解存在的一个一般性充分条件，然后构造相应的曲率流，并分析它们解的长时间存在性、渐近状态及奇异性等性质。最后，拟讨论这类方程解的唯一性问题，特别是某种特殊情形下解的唯一性问题。

Orlicz-Minkowski 问题是现代凸几何的 Orlicz-Brunn-Minkowski 理论中的首要基本问题之一，与此相关的一类 Monge-Ampere 型方程也具有许多重要的应用。但由于这类方程是超球面上的完全非线性偏微分方程，具有相当的难度和挑战性，并且它引起大家的关注也不过几年时间，因此有大量问题亟待解决。..本项目研究了 Orlicz-Minkowski 问题及相关 Monge-Ampere 型方程解的存在性、唯一性、奇异性等方面。具体地说，我们对 Orlicz-Minkowski 问题得到了一种新的测度条件下的解存在的充分性条件；对中心仿射 Minkowski 问题的高维旋转对称情形和平面上的一般情形证明了新的解存在性结果；我们得到了对偶 Orlicz–Minkowski 问题解的存在性和唯一性结果；通过构造证明了Lp 对偶 Minkowski 问题解的非唯一性；我们还研究了 Orlicz–Christoffel–Minkowski 问题并得到了解的存在性结果。..本项目在专业数学期刊上已正式发表学术论文9篇，包括《Advances in Mathematics》、《Transactions of the American Mathematical Society》、《International Mathematics Research Notices》、《Journal of Differential Equations》、《Journal of Geometric Analysis》、《Science China. Mathematics》等。