Lp-Minkowski 问题及相关的 Monge-Ampere 型方程

The Lp-Minkowski problem is an essential extension of the Minkowski problem which is very famous in geometry. It is a basic primary problem in the Lp-Brunn-Minkowski theory about convex geometry. The Lp-Minkowski problem and associated Monge-Ampere type equations have many important applications in other areas of mathematics. The Lp-Minkowski problem is studied by many mathematicians at home and abroad. For different values of p, the associated Monge-Ampere type equations have significantly different characteristics. Therefore there are still many problems about which people have not obtained satisfactory solutions. In particular, when p ≤ -n-1, the Lp-Minkowski problem corresponds to the critical or supercritical exponent of the famous Blaschke-Santalo inequality in convex geometry. It is very difficult and quite challenging, so very little is known about it. This project is intended to solve some important cases of the Lp-Minkowski problem. Specifically, we are firstly to find a general sufficient condition for the existence of solutions to the Lp-Minkowski problem when p = -n-1. Then we consider the associated parabolic equation, analyzing some properties of its solutions, such as long time existence, asymptotic behavior, singularity and so on. Based on these, we will study the size of the set of solvable functions. Finally we expect to describe all solutions to a special case of the Lp-Minkowski problem when p < -n-1.

Lp-Minkowski 问题是几何中著名的 Minkowski 问题的本质推广，是凸几何的“Lp-Brunn-Minkowski 理论”中首要的基本问题之一，与此相关的 Monge-Ampere 型方程在数学的其它领域里有重要应用。它虽然受到国内外众多学者的研究，但仍有许多问题还没有取得令人满意的结果。特别在 p ≤ -n-1 时，它对应于凸几何中著名的 Blaschke-Santalo 不等式的临界与超临界指数情形，具有相当的难度和挑战性，现有的结论更少。本项目拟在解决 Lp-Minkowski 问题中的某些重要情形。具体地说，我们拟首先寻找 p = -n-1 时 Lp-Minkowski 问题解存在的一个一般性充分条件，接着分析相应的抛物方程解的长时间存在性、渐近状态及奇异性等性质，然后在此基础上研究可求解函数集的大小。最后，拟描述 p < -n-1 时一种特殊情形下方程的解。

Lp-Minkowski 问题是几何中著名的 Minkowski 问题的本质推广，是凸几何的“Lp-Brunn-Minkowski 理论”中首要的基本问题之一，与此相关的 Monge-Ampere 型方程在数学的其它领域里有重要应用。它虽然受到国内外众多学者的研究，但仍有许多问题还没有取得令人满意的结果。特别在 p≤-n-1时，它对应于凸几何中著名的 Blaschke-Santalo 不等式的临界与超临界指数情形，具有相当的难度和挑战性，现有的结论更少。..本项目研究 Lp-Minkowski问题中的某些重要情形。具体地说，我们对 Lp-Minkowski 问题的次临界情形，构造了一个例子，说明这个方程在一般条件下其解不唯一；我们对 Lp-Minkowski 问题的临界情形，研究了相应的变分泛函，证明了其上确界无法取得；我们研究了临界情形的解存在性，对一些特殊情形找到了解存在的若干充分条件；我们研究了关于位置的预定 Lp-Gauss 曲率问题，得到了解存在的一个充分条件。..本项目的研究结果目前已形成了9篇论文，其中有6篇已在专业数学期刊《Advances in Mathematics》、 《Journal of Functional Analysis》、 《Calculus of Variations and Partial Differential Equations》、 《Journal of Differential Equations》、 《Discrete and Continuous Dynamical Systems》、 《Science China. Mathematics》上发表或被接收。