  We consider a partially hinged rectangular plate and its normal modes. The
dynamical properties of the plate are influenced by the spectrum of the
associated eingenvalue problem. In order to improve the stability of the plate,
it seems reasonable to place a certain amount of stiffening material in
appropriate regions. If we look at the partial differential equation appearing
in the model, this corresponds to insert a suitable weight coefficient inside
the equation. A possible way to locate such regions is to study the eigenvalue
problem associated to the aforementioned weighted equation. In the present
paper we focus our attention essentially on the first eigenvalue and on its
minimization in terms of the weight. We prove the existence of minimizing
weights inside special classes and we try to describe them together with the
corresponding eigenfunctions.
