  In this paper we analyze a shape optimization problem, with Stokes equations
as the state problem, defined on a domain with a part of the boundary that is
described as the graph of the control function. The state problem formulation
is mapped onto a reference domain, which is independent of the control
function, and the analysis is mainly led on such domain. The existence of an
optimal control function is proved, and optimality conditions are derived.
After the analytical inspection of the problem, finite element discretization
is considered for both the control function and the state variables, and a
priori convergence error estimates are derived. Numerical experiments assess
the validity of the theoretical results.
