“Biharmonic eigenvalues on annuli and rectangles”
Mercoledì 30 Ottobre 2019, ore 14:30 - Aula 2BC30 - Davide Buoso (EPFL, CH)
In this talk we discuss some nodal properties and shape optimization results for the eigenvalues of the Dirichlet Bilaplacian and of the Buckling problem. In particular, we focus our attention on two types of planar domains: annuli and rectangles. Annuli are good to study the nodal domains of eigenfunctions because they can be written explicitly in terms of Bessel functions, and we show that, quite unexpectedly, the two problems have completely different behaviors in terms of nodal domains. However, in both cases the annulus (of given area) that realizes the minimum first eigenvalue can be shown to be the ball in both cases. On the other hand, the situation for rectangles is reversed: any eigenfunction will present oscillations near the corners, but the shape optimization problem is highly non-trivial and we conjecture that it is solved by the square. We will also compare these results with the analogs for the Dirichlet Laplacian to show how the needed techniques change when passing from second-order problems to higher-order ones.
Based on joint works with P. Freitas and E. Parini.