Convergence of the solutions of the discounted equation

TL;DR

This paper proves the uniform convergence of solutions to the discounted Hamilton-Jacobi equation on compact manifolds as the discount factor approaches zero, identifying the limit in terms of Mather measures and Peierls barrier.

Contribution

It establishes the convergence of discounted solutions to a specific critical solution and characterizes this limit using variational tools.

Findings

Uniform convergence of $u_\lambda$ to $u_0$ as $\lambda o 0$

Characterization of the limit solution via Peierls barrier

Identification of the limit in terms of projected Mather measures

Abstract

We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_\lambda:M\to\mathbb R$ is the viscosity solution of the discounted equation $$ \lambda u_\lambda(x)+H(x,d_x u_\lambda)=c(H), $$ where $c(H)$ is the critical value, we prove that $u_\lambda$ converges uniformly, as $\lambda\to 0$, to a specific solution $u_0:M\to\mathbb R$ of the critical equation $$ H(x,d_x u)=c(H). $$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.