A quantitative rigidity result for a two-dimensional Frenkel-Kontorova model

TL;DR

This paper provides a quantitative estimate of equilibrium configurations in a 2D Frenkel-Kontorova model using PDE techniques and complex analysis, advancing understanding of lattice systems.

Contribution

It introduces a novel quantitative rigidity estimate for 2D Frenkel-Kontorova systems, combining PDE methods with complex variable techniques.

Findings

Quantitative estimate on the angular function of equilibria

Application of PDE methods inspired by De Giorgi's conjecture

Use of complex variables to analyze discrete lattice models

Abstract

We consider a Frenkel-Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by E. De Giorgi, also in view of an elegant technique based on complex variables that was introduced by A. Farina. In the discrete setting, a careful analysis of the reminders is needed to exploit this type of methodologies inspired by continuum models.