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

Let be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let be a semi-concave function. If is the set of all the super-differentials of u and (φ_t) the Hamiltonian flow of H, we prove that for t > 0 small enough, 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 C^1,1 subsolutions.