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

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.