Weak KAM theorem for Hamilton-Jacobi equations

TL;DR

This paper extends the weak KAM theorem to a broader class of Hamilton-Jacobi equations by introducing a solution semigroup and proving its convergence and the existence of weak KAM solutions.

Contribution

It generalizes the weak KAM theorem from positive Lagrangian systems to proper Hamilton-Jacobi equations using a new solution semigroup approach.

Findings

Proves convergence of the solution semigroup.

Establishes existence of weak KAM solutions for stationary equations.

Introduces an implicit solution semigroup for evolutionary Hamilton-Jacobi equations.

Abstract

In this paper, we generalize weak KAM theorem from positive Lagrangian systems to "proper" Hamilton-Jacobi equations. We introduce an implicitly defined solution semigroup of evolutionary Hamilton-Jacobi equations. By exploring the properties of the solution semigroup, we prove the convergence of solution semigroup and existence of weak KAM solutions for stationary equations: \begin{equation*} H(x, u, d_x u)=0. \end{equation*}