Convergence of discrete Aubry-Mather model in the continuous limit
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en
Article de revue
Este ítem está publicado en
Nonlinearity. 2018, vol. 31, n° 5
IOP Publishing
Resumen en inglés
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 ...Leer más >
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.< Leer menos
Palabras clave en inglés
discrete weak KAM theory
Frenkel-Kontorova models
AubryMather theory
discounted Lax-Oleinik operator
ergodic cell equation
shortrange interactions
additive eigenvalue problem
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