Weak KAM from a PDE point of view: viscosity solutions of the Hamilton-Jacobi equation and Aubry set (Q2866664)

In this paper, the author introduces the notion of a viscosity solution for the first-order Hamilton-Jacobi equation, in the more general setting of manifolds, to obtain a weak KAM theory using only tools from partial differential equations. This very pedagogical work is self-contained, and is aimed to people with no prior knowledge of the subject.NEWLINENEWLINEThe article is divided into 14 sections: Section 1: Different formulations of the Hamilton-Jacobi are given. Section 2: The notion of viscosity solution (resp. supersolution and subsolution) is presented. The author shows that a \(C^1\)-viscosity solution is classical (Theorem 2.4). Section 3: The notions of lower and upper differentials are introduced and studied. Section 4: Characterisations of viscosity solutions or given. Section 5: The author studies the case of coercive Hamiltonians. Section 6: A stability result for viscosity solutions is proven. Section 7: Uniqueness results are obtained. Section 8: The author introduces the Perron method for the construction of viscosity solutions. Section 9: The notion of strict subsolutions is discussed. Section 10: The case when the Hamiltonian is quasi-convex is studied. Section 11: The viscosity semi-distance is defined and studied. Section 12: The author defines and deals with the Aubry set. Section 13: A representation formula of a viscosity solution is proven. Section 14: The relationship between viscosity solutions, weak KAM solutions and the Lax-Oleinik semi-group for a Tonelli Hamiltonian is discussed.