Essential forward weak KAM solution for the convex Hamilton-Jacobi equation

TL;DR

This paper constructs a unique forward weak KAM solution for convex Hamilton-Jacobi equations on compact manifolds using a vanishing discount method, highlighting its dynamical importance.

Contribution

It introduces a novel approach to obtain the unique forward weak KAM solution via vanishing discount on compact manifolds.

Findings

Existence of a unique forward weak KAM solution.

Connection between the solution and dynamical systems.

Method applicable to convex, coercive Hamiltonians.

Abstract

For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$, we construct a unique forward weak KAM solution of \[ H(x, d_x u)=c(H) \] by a vanishing discount approach, where $c(H)$ is the Ma\~n\'e critical value. We also discuss the dynamical significance of such a special solution.