Maximization of the first eigenvalue in problems involving the bi-Laplacian

This paper concerns maximization of the first eigenvalue in problems involving the bi-Laplacian under either Navier boundary conditions or Dirichlet boundary conditions. Physically, in the case of N = 2, our equation models the vibration of a nonhomogeneous plate Ω which is either hinged or clamped along the boundary. Given several materials (with different densities) of total extension | Ω |, we investigate the location of these materials throughout Ω so as to maximize the first eigenvalue in the vibration of the corresponding plate.