Convergence of the solutions of the nonlinear discounted Hamilton-Jacobi equation: The central role of Mather measures

TL;DR

This paper investigates the convergence of solutions to nonlinear discounted Hamilton-Jacobi equations on manifolds, emphasizing the critical role of Mather measures and establishing conditions for existence, uniqueness, and convergence of solutions.

Contribution

It introduces new degeneracy conditions related to Mather measures that ensure convergence of solutions in nonlinear discounted Hamilton-Jacobi equations.

Findings

Solutions exist and are unique under specified conditions.

The family of solutions converges to a specific solution as the discount factor approaches zero.

Degeneracy conditions involving Mather measures are crucial for convergence.

Abstract

Given a continuous Hamiltonian $H : (x,p,u) \mapsto H(x,p,u)$ defined on $ T^*M \times \mathbb R $, where $M$ is a closed connected manifold, we study viscosity solutions, $u_\lambda : M\to \mathbb R$, of discounted equations: $ H(x, d_x u_\lambda, \lambda u_\lambda(x))=c$ in $M$, where $\lambda >0$ is called a discount factor and $c$ is the critical value of $H(\cdot, \cdot , 0)$. When $H$ is convex and superlinear in $p$ and non--decreasing in $u$, under an additional non--degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions $(u_\lambda)_{\lambda >0}$ converges to a specific solution $u_0$ of $ H(x, d_x u_0, 0)=c$ in $M$. Our degeneracy condition requires $H$ to be increasing (in $u$) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in $u$.