  In this paper we deal with the composite plate problem, namely the following
optimization eigenvalue problem $$ \inf_{\rho \in \mathrm{P}} \inf_{u \in
\mathcal{W}\setminus\{0\}} \frac{\int_{\Omega}(\Delta u)^2}{\int_{\Omega} \rho
u^2}, $$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}=
H^{2}_{0}(\Omega)$ for Dirichlet boundary conditions and $\mathcal W=
H^2(\Omega) \cap H^1_{0}(\Omega)$ for Navier boundary conditions. The
associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by
the biharmonic operator $\Delta^2$. In the spirit of [10], we study qualitative
properties of the optimal pairs $(u,\rho)$. In particular, we prove existence
and regularity and we find the explicit expression of $\rho$. When $\Omega$ is
a ball, we can also prove uniqueness of the optimal pair, as well as positivity
of $u$ and radial symmetry of both $u$ and $\rho$.
