Convergence of discrete Aubry-Mather model in the continuous limit

• Published in: Nonlinearity Vol. 31; no. 5; pp. 2126 - 2155
• Main Authors: Su, Xifeng, Thieullen, Philippe
• Format: Journal Article
• Language: English
• Published: IOP Publishing 06.04.2018
• Subjects: additive eigenvalue problem; Analysis of PDEs; Aubry-Mather theory; discounted Lax-Oleinik operator; discrete weak KAM theory; Dynamical Systems; ergodic cell equation; Frenkel-Kontorova models; Mathematics; short-range interactions; discounted Lax-Oleinik operator; discrete weak KAM theory; Frenkel-Kontorova models; additive eigenvalue problem; shortrange interactions; AubryMather theory; ergodic cell equation
• ISSN: 0951-7715; 1361-6544; 1361-6544
• DOI: 10.1088/1361-6544/aaacbb

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 (2016 Invent. Math. 206 29-55), and show that 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 the short-range interactions.