MATHER THEORY, WEAK KAM, AND VISCOSITY SOLUTIONS OF HAMILTON-JACOBI PDE’S

standard n-dimensional torus and Rn is the n-dimensional Euclidean space, r ≥ 3. Denote coordinates x = (x1,..., xn) ∈ Tn and p = (p1,..., pn) ∈ Rn. Call x – position. We shall investigate dynamics of the Hamiltonian equations:{ x ̇ = ∂pH(x, p), p ̇ = −∂xH(x, p). (1) We call the following three assumptions standard assumptions • H is convex in p, i.e. for all x ∈ Tn we have that the Hessian matrix ∂2pipjH(x, p) is positive definite for all p ∈ Rn. • H is superlinear in p, i.e. for all x ∈ Tn we have that limH(x, p)/|p | → +∞ as |p | → +∞ • The flow defined by (1) is complete, i.e. for each initial condition (x0, p0) ∈ Tn × Rn solutions of (1) exists for all time. Actually weak KAM and Mather theories we describe below work for a Hamil-tonian 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) — C2 smooth potential on Tn. 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) ∈ Rn a lift of x(t) ∈ Tn to the universal cover Rn over Tn = Rn/Tn. 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. lim |t|→+∞ x̂(t) − x̂(0