PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians
The main purpose of this paper is to generalize the Aubry-Mather theory for Hamiltonians \(H\), defined on the cotangent bundle of the flat torus \(\mathbb T^N\), satisfying weak regularity assumptions. More precisely, in the first part of the paper \(H\) is coercive quasiconvex and continuous, and in the second part a strict convexity as well as a local Lipschitz-continuity assumption is added. In such a setting, no Hamiltonian dynamics can necessarily be defined, at least in the usual sense, and consequently dynamical systems techniques are not suitable. The authors attack the problem using PDE methods, and in particular the theory of viscosity solutions. Then, the family of Hamilton-Jacobi equations \(H(x,Du)= a\) on \(\mathbb T^N\), with \(a\) a real parameter, is considered. It is well known that a special value of \(a\) is relevant, qualified as critical and denoted in the remainder by \(c\). It is characterized, from a PDE viewpoint, by the property that the corresponding equation can be solved in the viscosity sense on the whole torus. Finally, a generalization of the projected Mather set, under the restriction that the set of equilibria is empty, is provided and it is proved that it is a uniqueness set for the equation \(H(x, Du)= c\).
scientific article; zbMATH DE number 2135760
scientific article