Optimization of eigenvalues of partially hinged rectangular composite plates

We study the spectrum of non-homogeneous partially hinged rectangular plates having structural engineering applications. A possible way to prevent instability phenomena is to optimize the frequencies of certain oscillating modes with respect to the density function of the plate. In striking contrast to what happens under Dirichlet boundary conditions, we prove a result of positivity preserving property for the biharmonic operator of the related problem through fine estimates of the Fourier expansion of the corresponding Green function. This is useful in order to get qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue.
This is a joint work with Elvise Berchio (Politecnico di Torino).