Maximization and minimization in problems involving the bi-Laplacian

This paper concerns minimization and maximization of the energy integral in problems involving the bi-Laplacian under either homogeneous Navier boundary conditions or homogeneous Dirichlet boundary conditions. Physically, in case of N = 2, our equation models the equilibrium configuration of a non-homogeneous 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 inside Ω so to maximize or minimize the energy integral of the corresponding plate.