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

Ahmad Fino (1, 2, 3) , Hassan Ibrahim (3, 4, 5) , Régis Monneau (4)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 MIA - Mathématiques, Image et Applications - EA 3165
3 LaMA--Liban - Laboratoire de mathématiques et applications
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
5 LU / ULB - الجامعة اللبنانية [بيروت] = Lebanese University [Beirut] = Université libanaise [Beyrouth]

We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ 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.

Mots clés: Frenkel-Kontorova; Peierls-Nabarro; viscosity solutions; dislocations; discrete-continuous; boundary conditions

Domaines: Equations aux dérivées partielles [math.AP]

Ahmad Fino, Hassan Ibrahim, Régis Monneau. The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model. 2010. ⟨hal-00523488⟩