  A Bernoulli free boundary problem with geometrical constraints is studied.
The domain $\Om$ is constrained to lie in the half space determined by $x_1\geq
0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where
non-homogeneous Dirichlet conditions are imposed. We are then looking for the
solution of a partial differential equation satisfying a Dirichlet and a
Neumann boundary condition simultaneously on the free boundary. The existence
and uniqueness of a solution have already been addressed and this paper is
devoted first to the study of geometric and asymptotic properties of the
solution and then to the numerical treatment of the problem using a shape
optimization formulation. The major difficulty and originality of this paper
lies in the treatment of the geometric constraints.
