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

TL;DR

This paper provides a geometric and dynamical interpretation of the Lax-Oleinik semi-group using pseudographs and Hamiltonian flows, enhancing understanding of regularization in Hamilton-Jacobi equations.

Contribution

It introduces a new geometric perspective on the Lax-Oleinik semi-group through pseudographs and Hamiltonian dynamics, connecting regularization to Lagrangian Lipschitz graphs.

Findings

For small t, the Hamiltonian flow maps super-differentials to Lagrangian Lipschitz graphs.

Provides a geometric explanation for a regularization tool used in Hamilton-Jacobi theory.

Enhances understanding of the structure of solutions to Hamilton-Jacobi equations.

Abstract

Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u be a semi-concave function defined on M. If E (u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian flow of H, we prove that for t > 0 small enough, \phi-t (E (u)) is an exact Lagrangian Lipschitz graph. This provides a geometric interpretation/explanation of a regularization tool that was introduced by P.~Bernard to prove the existence of C 1,1 subsolutions.