凸几何中蒙日-安培型方程的研究

近几年，随着对偶曲率测度的提出，对偶Minkowski型问题理论不断发展，并引起人们广泛关注。对偶Minkowski型问题是凸体中经典的Brunn-Minkowski理论的发展，对它们的研究通常转化为解一类球面上的蒙日-安培型方程。Lp对偶Minkowski问题在众多学者的研究下取得许多重要进展，但仍有关键问题没有满意的结果，特别是在有孤立奇点的情况下无任何结果。对偶Orlicz-Minkowski问题是更加困难的问题，加之问题的提出时间不长，许多关键问题亟待解决。.本项目拟研究上述对偶Minkowski型问题及相关的蒙日-安培型方程中的关键问题。我们具体研究Lp对偶Minkowski问题解的唯一性和对偶Orlicz-Minkowski问题及Orlicz-Minkowski问题解的存在性和唯一性以及它们具有孤立奇点的情况。同时，我们将深入研究更广泛的一类各项异性曲率流的性态。

Rencently, dual Minkowski problems have attracted more attentions since the dual curvature measure was put forward. Dual Minkowski problems are the extensions of classical Brunn-Minkowski theory in convex geometry, which generally are reduced to Monge-Ampere type equations on unit sphere. Though important progress was made on Lp dual Minkowski problem by many mathematicians, some crucial problems are still open, in particular, when an isolated singular point appears. Dual Orlicz-Minkowski problem is more difficult. Since the dual Orlicz-Minkowski problem has been only studied for a few years, there are many important problems to be solved. In this project, we intend to study Lp dual Minkowski problem and dual Orlicz-Minkowski problem and related Monge-Ampere type equations. We will study the uniqueness and existence of Lp dual Minkowski problem and the sufficient conditions for the uniqueness and existence of dual Orlicz-Minkowski problem and Orlicz-Minkowski problem, including the case in which an isolated singular point appears. In the same time, we will study the properties of a generalized anisotropic curvature flow.