# The Hamilton-Jacobi Equation and Weak KAM Theory

# The Hamilton-Jacobi Equation and Weak KAM Theory

This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.

*Keywords:*
invariant sets, KAM theory, Albert Fathi, MAK tori, Hamilton–Jacobi equation, Aubry–Mather theory