Solutions of Neumann problems in domains with cracks and applications to fracture mechanics
Gianni Dal Maso · arXiv (Cornell University) · 2001
The first part of the course is devoted to the study of solutions to the Laplace equation in $Ω\setminus K$, where $Ω$ is a two-dimensional smooth domain and $K$ is a compact one-dimensional subset of $Ω$. The solutions are required to satisfy a homogeneous Neumann boundary condition on $K$ and a nonhomogeneous Dirichlet condition on (part of) $\partialΩ$. 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.