Symmetry and rigidity results for composite membranes and plates

The composite membrane problem is an eigenvalue optimization problem deeply studied from the beginning of the '00's. In this note we survey most of the results proved by several authors over the last twenty years, up to the recent paper [14] written in collaboration with Giovanni Cupini.We finally introduce an eigenvalue optimization problem for a fourth order operator, called composite plate problem and we present the symmetry and rigidity results obtained in this framework. These last mentioned results are part of the papers [12,13], written in collaboration with Francesca Colasuonno.