Symmetry in the composite plate problem

In this paper, we deal with the composite plate problem, namely the following optimization eigenvalue problem: equation presented where P is a class of admissible densities, W = H 20 (Ω) for Dirichlet boundary conditions and W = H 2 (Ω) ∩ H 10 (Ω) for Navier boundary conditions. The associated Euler- Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator Δ 2 . In the spirit of [S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys. 214 (2000) 315-337], we study qualitative properties of the optimal pairs (u, ρ). In particular, we prove existence and regularity and we find the explicit expression of ρ.When Ω 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 ρ.