Consider a Hamiltonian system with Hamiltonian of the form H (cursive Greek chi, t, p) where H is convex in p and periodic in cursive Greek chi, and t and cursive Greek chi ∈ double-struck R sign1. It is well-known that its smooth invariant curves correspond to smooth Z2-periodic solutions of the PDE ut + H (cursive Greek chi, t, u)cursive Greek chi =0. In this paper, we establish a connection between the Aubry-Mather theory of invariant sets of the Hamiltonian system and Z2-periodic weak solutions of this PDE by realizing the Aubry-Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry-Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry-Mather set, defined in (2.24). The graph itself is a backward-invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry-Mather theory into the characteristic fields of the above PDE. This is done by making use of one- and two-sided minimizers, a notion introduced in  and inspired by the work of Morse on geodesics of type A , The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in . As an application, we prove that the Z2-periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two-sided minimizers with a specified asymptotic slope may not exist.
|Original language||English (US)|
|Number of pages||18|
|Journal||Communications on Pure and Applied Mathematics|
|State||Published - Jul 1 1999|
All Science Journal Classification (ASJC) codes
- Applied Mathematics