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

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