Convergence of discrete Aubry-Mather model in the continuous limit
This provides a rigorous foundation for numerical approximation of weak KAM solutions, relevant to researchers in dynamical systems and PDEs.
The authors develop discrete approximation schemes for solving cell equations and discounted cell equations from Aubry-Mather theory, proving convergence of discrete weak KAM solutions to continuous ones as the time step tends to zero.
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.