Mathematics – Analysis of PDEs
Scientific paper
2001-05-16
Mathematics
Analysis of PDEs
Lecture notes of a course held in the 2001 CNA Summer School ``Multiscale Problems in Nonlinear Analysis'', Carnegie Mellon Un
Scientific paper
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.
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