  We give a precise mathematical formulation of a variational model for the
irreversible quasi-static evolution of brittle fractures proposed by G.A.
Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In
the two-dimensional case we prove an existence result for the quasi-static
evolution and show that the total energy is an absolutely continuous function
of time, although we can not exclude that the bulk energy and the surface
energy may present some jump discontinuities. This existence result is proved
by a time discretization process, where at each step a global energy
minimization is performed, with the constraint that the new crack contains all
cracks formed at the previous time steps. This procedure provides an effective
way to approximate the continuous time evolution.
