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 these types of methodologies inspired by continuum models.

中文翻译：

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二维Frenkel-Kontorova模型的定量刚度结果

我们考虑了二维欧几里得格子中的Frenkel-Kontorova谐振子系统，并获得了关于平衡角函数的定量估计。该证明依赖于与E. De Giorgi的经典猜想相关的PDE方法，同时考虑到A. Farina引入的基于复杂变量的优雅技术。

在离散环境中，需要对提醒进行仔细的分析，以利用由连续模型产生的这些类型的方法。