Symmetry in the composite plate problem

Abstract

In this paper, we deal with the composite plate problem, namely the following optimization eigenvalue problem: [Formula: see text] where [Formula: see text] is a class of admissible densities, [Formula: see text] for Dirichlet boundary conditions and [Formula: see text] for Navier boundary conditions. The associated Euler–Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator [Formula: see text]. 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 [Formula: see text]. In particular, we prove existence and regularity and we find the explicit expression of [Formula: see text]. When [Formula: see text] is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of [Formula: see text] and radial symmetry of both [Formula: see text] and [Formula: see text].