闵可夫斯基空间中极值曲面的研究

2002 年Brenier 教授在国际数学家大会报告中阐述了闵可夫斯基空间极值曲面及相关物理模型在研究几何学和粒子物理中的重要意义。本项目将主要研究闵可夫斯基空间中的时向极值曲面行波解的稳定性、高维外问题及混合型极值曲面解的整体适定性等有重要物理和理论意义的数学问题。特别地，我们将主要研究如下内容：一、在一维空间情形：1)研究时向极值曲面方程柯西问题和混合初边值问题行波解的稳定性；2)研究混合型极值曲面带混合型初值问题的整体解存在性；3)粒子物理中重要的Faddeev 模型柯西问题行波解的存在性与稳定性。二、在以上基础上，力争对(1+2+1)维情形的时向极值曲面在任意时向平面内的星型区域外问题的整体经典解这一重要公开问题取得进展或者突破。

The important of the extremal suface in Minkowski space was elaborated by Prof. Brenier on his talk in 2002 ICM (International Congress of Mathematicians). This project contains the stability of traveling wave solutions, global solutions to the mixed type extremal surface and high dimensional timelike extremal surface in exterior domain and related physical models. Therefore, this research will provide the theoretical foundation for the geometrical surface theory and particle physics...Based on our obtained works about the asymptotic behavior and global existence of the classical solutions of Cauchy problem and the mixed initial-boundary value problem for the time-like extremal surface in Minkowski space, we will consider the following issues:.Case 1: in one space dimensional Minkowski space.1) The stability of traveling wave solutions of Cauchy problem (initial-boundary value problem) to the equation of time-like extremal surface..2) The global existence of classical solutions to the mixed type extremal surface in Minkowski space with the mixed initial data..3) The stability of travelling wave solutions to Cauchy problem of Faddeev model in particle physics..Case 2: in (1+2+1) dimensional Minkowski space.The global existence of classical solutions to the time-like extremal surface in exterior domains with homogeneous (or inhomogeneous) boundary conditions which is one of open problems in the field of nonlinear wave equations.