Ghost circles in lattice Aubry-Mather theory

B. Mramor, B.W. Rink

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

Monotone lattice recurrence relations such as the Frenkel-Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps.Such recurrence relations often admit a variational structure, so that the solutions x:Zd→R are the stationary points of a formal action function W(x). Given any rotation vector ω∈Rd, classical Aubry-Mather theory establishes the existence of a large collection of solutions of ∇. W(x) = 0 of rotation vector ω. For irrational ω, this is the well-known Aubry-Mather set. It consists of global minimizers and it may have gaps.In this paper, we study the parabolic gradient flow dxdt=-∇;W(x) and we will prove that every Aubry-Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called 'ghost circle'. The existence of these ghost circles is known in dimension d=1, for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions.As a consequence, we can give a simple proof of the fact that when an Aubry-Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution. © 2011 Elsevier Inc.
Original languageEnglish
Pages (from-to)3163-3208
JournalJournal of Differential Equations
Volume252
Issue number4
DOIs
Publication statusPublished - 2012

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