Smooth critical sub-solutions of the Hamilton-Jacobi equation

We establish the existence of smooth critical sub-solutions of the HamiltonJacobi equation on compact manifolds for smooth convex Hamiltonians, that is in the context of weak KAM theory, under the assumption that the Aubry set is the union of finitely many hyperbolic periodic orbits or fixed points. Let M be a compact manifold without boundary. A function H(x, p) : T M −→ R is called a Tonelli Hamiltonian if it is C and if, for each x ∈ M the function p 7−→ H(x, p) is convex with positive definite Hessian and superlinear on the fibre T ∗ xM . A Tonelli Hamiltonian generates a complete Hamiltonian flow ψt. It is known that there exists one and only one real constant α(H) such that the equation (HJ) H(x, dux) = α(H) has a solution in the viscosity sense. This equation, then, may have several solutions. These solutions are Lipschitz but, in general, none of them is C. It is a natural question whether there exist sub-solutions of (HJ) which are more regular. In this direction, Fathi and Siconolfi proved the existence of a C sub-solution, see [14]. More recently, I proved the existence of C sub-solutions see [4]. Examples show that, in general, C sub-solutions do not exist. In the present paper, we establish, under the additional assumption that the Aubry set is a finite union of hyperbolic orbits and hyperbolic fixed points, the existence of smooth sub-solutions. This answers a question asked several times to the author and to Albert Fathi during various conferences on weak KAM theory. This also provides a nice class of examples, and is a useful technical step for deeper studies of this class of examples, see [1] and other papers.