Symmetry in the composite plate problem

In this paper we deal with the {it composite plate problem}, namely the following optimization eigenvalue problem $$ inf_{ ho in mathrm{P}} inf_{u in mathcal{W}setminus{0}} rac{int_{Omega}(Delta u)^2}{int_{Omega} ho u^2}, $$ 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 cite{CGIKO00}, we study qualitative properties of the optimal pairs $(u, ho)$. In particular, we prove existence and regularity and we find the explicit expression of $ ho$. 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 $ ho$.