Regularity of weak KAM solutions and Man˜e's Conjecture L. Rifford? Abstract We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Man˜e conjecture. 1 Introduction In the present paper, (M, g) will be a smooth connected compact Riemannian manifold without boundary of dimension n ≥ 2, and H : T ?M ? R a Ck Tonelli Hamiltonian (with k ≥ 2), that is, a Hamiltonian of class Ck satisfying the two following properties (? · ? denotes the dual norm on T ?M): (H1) Superlinear growth: For every K ≥ 0, there is a finite constant C?(K) such that H(x, p) ≥ K?p?x + C ?(K) ? (x, p) ? T ?M. (H2) Uniform convexity: For every (x, p) ? T ?M , the second derivative along the fibers ∂2H ∂p2 (x, p) is positive definite. The Man˜e critical value of H can be defined as follows. Definition 1.1. We call critical value of H, denoted by c[H], the infimum of the values c ? R for which there exists a function u : M ? R of class C1 satisfying H(x, du(x)) ≤ c ?x ?M.
- ttu
- following properties
- arzela-ascoli theorem
- lax-oleinik semigroup
- inf z?m
- supremum norm
- compact riemannian
- fenchel inequality
- lipschitz curve
- legendre-fenchel duality