弱KAM理论和Hamilton-Jacobi方程

Weak KAM theory and Hamilton-Jacobi equations

Mather理论研究了在高维正定Lagrangian系统里各类作用量极小集的存在性以及适当条件下,这些作用量极小集之间的连接轨道的存在性,其中关于连接轨道的工作在Arnold扩散的研究中起着重要的作用.Fathi A.创立的弱KAM理论通过研究作用量极小曲线的动力学行为,在Mather理论及传统研究Hamilton-Jacobi所采用的PDE方法中建立起了桥梁.但由于在弱KAM理论中起核心作用的Lax-Oleinik半群在时间周期系统中的非收敛性,使得弱KAM理论的前期工作集中于自治系统.通过新型Lax-Oleinik算子的引入,使得在时间周期Lagrange系统建立弱KAM理论成为可能,也使得我们可能将弱KAM理论推广至更一般的Hamilton-Jacobi方程.本文我们介绍弱KAM理论以及有关这方面研究的最新进展.

Mather theory studies the existence of various kinds of the action minimizing sets and the connecting orbits between these sets in higher-dimensional positive Lagrangian system. The work about the connecting orbits plays an important role in the study of Arnold diffusion. By studying the dynamical behavior of the action-minimizing curves for Tonelli Lagrangian systems, weak KAM theory founded by A. Fathi bridges Mather theory and the PDE methods concerning the associated Hamilton-Jacobi equation. However, because the convergence of the Lax-Oleinik semigroup which is critical in KAM theory does not hold in time-periodic Lagrangian system, the preliminary work about weak KAM theory focused on autonomous system. By introducing a new kind of Lax-Oleinik type operator, it is possible for us to build weak KAM theory in the time-periodic Lagrangian system and generalize the theory to more general Hamilton-Jacobi equation. In this paper, we introduce the basic knowledge of weak KAM theory and its latest development.