  A shape optimization problem arising from the optimal reinforcement of a
membrane by means of one-dimensional stiffeners or from the fastest cooling of
a two-dimensional object by means of ``conducting wires'' is considered. The
criterion we consider is the maximization of the first eigenvalue and the
admissible classes of choices are the one of one-dimensional sets with
prescribed total length, or the one where the constraint of being connected (or
with an a priori bounded number of connected components) is added. The
corresponding relaxed problems and the related existence results are described.
