The Peierls–Nabarro model as a limit of a Frenkel–Kontorova model

We study a generalization of the fully overdamped Frenkel–Kontorova model in dimension n⩾1. This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls–Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions.