Pseudographs and the Lax–Oleinik semi-group: a geometric and dynamical interpretation

Let H: T*M→R be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u: M→R be a semi-concave function. If E(u) is the set of all the super-differentials of u and (ψt) the Hamiltonian flow of H, we prove that for t > 0 small enough, φ-t(E(u)) is an exact Lagrangian Lipschitz graph. This provides a geometric interpretation/explanation of a regularization tool that was introduced by Bernard (2007 Ann. Sci. École Norm. Sup. 40 445–52) to prove the existence of C1,1 subsolutions