Minimization of the first eigenvalue in problems involving the bi-laplacian

Abstract

This paper concerns the minimization of the first eigenvalue 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 vibration of a non homogeneous plate which is either hinged or clamped along the boundary. Given several materials (with different densities) of total extension | |, we investigate the location of these materials inside so to minimize the first mode in the vibration of the corresponding plate. Keywords: bi-Laplacian, first eigenvalue, minimization.