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\begin{center}{\bf Solutions of Neumann problems in domains with cracks and applications to fracture mechanics}
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The first part of the course deals with the study of solutions of
the Laplace equation in  $\Omega\setminus K$, where 
 $\Omega$ is
a two dimensional smooth domain and $K$ is a connected compact 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. Classical examples show that the result is no
longer true without the hypothesis that $K$ is connected.

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The second part of the course uses this stability result
to give a rigorous mathematical formulation of a variational quasistatic
model for the slow growth of a brittle fracture, 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 limit argument. Griffith's
criterion for fracture growth is stated and proved within this model.

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