Abstract
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.
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