Inserted: 16 may 2001
Last Updated: 18 may 2001
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
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.
Keywords: Free-discontinuity problems, quasi-static evolution, stability of Neumann problems, domains with cracks