弱KAM理论中的若干问题

由法国数学家A. Fathi于上世纪90年代中期创立的弱KAM理论是目前Hamilton动力系统研究领域中最为活跃的研究课题之一。鉴于它重要的理论和应用价值，弱KAM理论受到了众多国内外数学家的关注，得到了快速的发展。本项目将紧密围绕弱KAM理论的核心工具-Lax-Oleinik半群（或Lax-Oleinik型算子）对该理论中的三个问题进行细致的研究。这三个问题分别为：1. 自治Lagrange系统的Lax-Oleinik半群及其C1图形收敛速度与Aubry集动力学性质间的关系；2. 时间周期Lagrange系统的新Lax-Oleinik型算子的收敛速度与Aubry集动力学性质的关系；3. 构建随机情形弱KAM理论的基本框架。

The weak KAM theory founded by the French mathematician A. Fathi in the mid-1990s is currently one of the most active research topics in the domain of Hamiltonian dynamical systems. In view of its important theoretical and practical value,the weak KAM theory attracts more and more attention of domestic and foreign mathematicians,and has been developing rapidly. The project focuses on the core tool of the weak KAM theory-Lax-Oleinik semigroup (or Lax-Oleinik type operators), and discusses in detail three problems in the weak KAM theory: 1. the problem of the relationship between dynamical properties of the Aubry set and the rates of convergence of the Lax-Oleinik semigroup associated with time-independent Lagrangian systems and its C1 graph; 2. the problem of the relationship between dynamical properties of the Aubry set and the rate of convergence of the new Lax-Oleinik type operators associated with time-periodic Lagrangian systems; 3. the construction of the basic framework of a random analog of the weak KAM theory.