Mather Theory, Weak Kam, and Viscosity Solutions of Hamilton-jacobi Pde’s

We call the following three assumptions standard assumptions • H is convex in p, i.e. for all x ∈ T we have that the Hessian matrix ∂ pipjH(x, p) is positive definite for all p ∈ R. • H is superlinear in p, i.e. for all x ∈ T we have that lim H(x, p)/|p| → +∞ as |p| → +∞ • The flow defined by (1) is complete, i.e. for each initial condition (x0, p0) ∈ T × R solutions of (1) exists for all time. Actually weak KAM and Mather theories we describe below work for a Hamiltonian defined on any smooth compact manifold. We consider the case of a Hamiltonian on a torus for exposition purposes only. The Basic Example is the standard mechanical system: H(x, p) = 〈p,p〉 2 + V (x) with 〈·, ·〉 — Euclidean scalar product and V (x) — C smooth potential on T. Let {(x(t), p(t))}t∈R be a trajectory of (1) with an initial conditions (x(0), p(0)) = (x0, p0). Denote by x̂(t) ∈ R a lift of x(t) ∈ T to the universal cover R over T = R/T. Definition 1.1. (Position) rotation vector of a trajectory {(x(t), p(t))}t∈R (if exists) equals average asymptotic velocity of x, i.e.