  We present first a brief review of the existing literature on shape
optimization, stressing the recent use of Hamiltonian systems in topology
optimization. In the second section, we collect some preliminaries on the
implicit parametrization theorem, especially in dimension two, which is a case
of interest in shape optimization. The formulation of the problem is also
discussed. The approximation via penalization and its differentiability
properties are analyzed in Section 3. Next, we investigate the discretization
process in Section 4. The last section is devoted to numerical experiments.
