Workshop on Symplectic Dynamics

This is a joint work with Renato Iturriaga.

We consider a Hamiltonian $H: \mathbf R^n \times \mathbf R \to \mathbf R, \ (x,p) \mapsto H(x,p)$ that is $Z^n$ periodic in the first variable $x$, and convex superlinear in the second variable $p$.

For $\lambda > 0$ we consider (viscosity) solutions of the discounted Hamilton-Jacobi equation \[ \lambda u_\lambda + H(x,Du_\lambda (x)) =c[0],\] where $c[0]$ is the unique constant $c$ such that the stationary Hamilton-Jacobi equation \[ H(x,Du(x)) =c \] has a viscosity solution. It is well-known that $u_\lambda$ is unique and that $u_\lambda$ accumulates on viscosity solutions of the stationary Hamilton-Jacobi equation, when $\lambda \to 0$.

We address the problem of actual convergence of $u_\lambda$ when $\lambda \to 0$.Using weak KAM theory, we can show that $u_\lambda$ converges to a unique viscosity solution of the stationary Hamilton-Jacobi equation, provided the Mather quotient of the Aubry Mather set has measure $0$ for the $1$-dimensional Hausdorff measure.

In particular, this is the case when the Mather quotient of the Aubry set is at most countable (generic condition on $H$), and also when $n \leq 3$ and $H$ is $C^4$.

We will recall the elements from Aubry-Mather and weak KAM theory that are necessary to understand the lecture.