Prof. Antonio Gaudiello “A spectral problem for the Laplacian in joined thin films”

We consider a 3d multi-structure composed of two joined perpendicular thin films: a vertical one with small thickness h^a_n and a horizontal one with small thickness h^b_n. We study the asymptotic behavior, as h^a_n and h^b_n tend to zero, of an eigenvalue problem for the Laplacian defined on this multi-structure. We shall prove that the limit problem depends on the value q=\lim_n\dfrac{h^b_n}{h^a_n}. Precisely, we pinpoint three different limit regimes according to q belonging to ]0, +∞[, q equal to +∞, or q equal to 0. We identify the limit problems and we also obtain H^1-strong convergence results.

This work is in cooperation with Delfina Gómez and Maria-Eugenia Pérez-Martínez (Universidad de Cantabria, Santander, Spain).