Albert Fathi (ENS Lyon)

The interface of PDEs and Lagrangian dynamics: existence of C¹ subsolutions

Abstract. This is a joint work with Antonio Siconolfi (La Sapienza, Roma).

We will discuss for Lagrangian Dynamical Systems (satisfying the usual Mather's condition), the relevance of subsolutions of the corresponding Hamilton-Jacobi equation.

The main result is to show that there is a C¹ subsolution of u satisfying $H(x,d_xu)\le c[0]$, where c[0] is the Mañé critical value. This settles a ``calibration'' problem related to the Aubry(-Mather) set, that has been studied, among others, by Ricardo Mañé in his last papers. It has also consequences on the dynamics of the Lagrangian System.

The proof is via some deep connections of the Aubry(-Mather) set, and viscosity solutions of Hamilton-Jacobi equations. This connection will also explain why it is impossible to obtain C² subsolutions.