Convergence of discrete Aubry-Mather model in the continuous limit

TL;DR

This paper introduces two approximation schemes for solving Hamilton-Jacobi equations using Aubry-Mather theory, demonstrating convergence of discrete models to continuous solutions and unifying the frameworks through short-range interactions.

Contribution

It develops novel approximation methods for the cell and discounted cell equations, establishing convergence of discrete solutions to continuous weak KAM solutions.

Findings

Discrete models approximate effective Hamiltonian as time step tends to zero

Selected discrete weak KAM solutions converge to continuous solutions

Unified formalism for continuous and discrete Aubry-Mather models

Abstract

We develop two approximation schemes for solving the cell equation and the discounted cell equation using Aubry-Mather-Fathi theory. The Hamiltonian is supposed to be Tonelli, time-independent , and periodic in space. By Legendre transform it is equivalent to find a fixed point of some nonlinear operator, called Lax-Oleinik operator, which may be discounted or not. By discretizing in time, we are led to solve an additive eigenvalue problem involving a discrete Lax-Oleinik operator. We show how to approximate the effective Hamiltonian and some weak KAM solutions by letting the time step in the discrete model tend to zero. We also obtain a selected discrete weak KAM solution as in [Davini et al 2014] and show it converges to a particular solution of the cell equation. In order to unify the two settings, continuous and discrete , we develop a more general formalism of short-range interactions.