  This article is devoted to the study of spectral optimisation for
inhomogeneous plates. In particular, we optimise the first eigenvalue of a
vibrating plate with respect to its thickness and/or density. Our result is
threefold. First, we prove existence of an optimal thickness, using fine tools
hinging on topological properties of rearrangement classes. Second, in the case
of a circular plate, we provide a characterisation of this optimal thickness by
means of Talenti inequalities. Finally, we prove a stability result when
assuming that the thickness and the density of the plate are linearly related.
This proof relies on H-convergence tools applied to biharmonic operators.
