Weak KAM theory for a weakly coupled system of Hamilton-Jacobi equations

Here, we extend the weak KAM and Aubry-Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton-Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi's weak KAM theorem, and describe the asymptotic limit of the generalized Lax-Oleinik semigroup.