{
  "id": "1010.0865",
  "version": "v1",
  "published": "2010-10-05T12:31:14.000Z",
  "updated": "2010-10-05T12:31:14.000Z",
  "title": "The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model",
  "authors": [
    "Ahmad Fino",
    "Hassan Ibrahim",
    "Régis Monneau"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-05T12:31:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "peierls-nabarro model",
      "coupled first order odes",
      "viscosity solutions",
      "elliptic pde",
      "solution converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.0865F"
    }
  }
}