{
  "id": "math/0105132",
  "version": "v1",
  "published": "2001-05-16T15:24:35.000Z",
  "updated": "2001-05-16T15:24:35.000Z",
  "title": "Solutions of Neumann problems in domains with cracks and applications to fracture mechanics",
  "authors": [
    "Gianni Dal Maso"
  ],
  "comment": "Lecture notes of a course held in the 2001 CNA Summer School ``Multiscale Problems in Nonlinear Analysis'', Carnegie Mellon University, Pittsburgh, May 31--June 9, 2001; 15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The first part of the course is devoted to the study of solutions to the Laplace equation in $\\Omega\\setminus K$, where $\\Omega$ is a two-dimensional smooth domain and $K$ is a compact one-dimensional subset of $\\Omega$. The solutions are required to satisfy a homogeneous Neumann boundary condition on $K$ and a nonhomogeneous Dirichlet condition on (part of) $\\partial\\Omega$. The main result is the continuous dependence of the solution on $K$, with respect to the Hausdorff metric, provided that the number of connected components of $K$ remains bounded. Classical examples show that the result is no longer true without this hypothesis. Using this stability result, the second part of the course develops a rigorous mathematical formulation of a variational quasi-static model of the slow growth of brittle fractures, recently introduced by Francfort and Marigo. Starting from a discrete-time formulation, a more satisfactory continuous-time formulation is obtained, with full justification of the convergence arguments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-05-16T15:24:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "74R10",
      "49Q10",
      "35A35",
      "35B30",
      "35J25"
    ],
    "keywords": [
      "fracture mechanics",
      "neumann problems",
      "applications",
      "homogeneous neumann boundary condition",
      "two-dimensional smooth domain"
    ],
    "tags": [
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......5132D"
    }
  }
}