27:42
publiée le 21 novembre 2025
Local decay and asymptotic profile for the damped wave equation in the asymptotically Euclidean setting
De Rayan Fahs
Dimension reduction in thin domains - Lecture 2
Apparaît dans la collection : Wave propagation in guiding structures / Propagation d'ondes dans les structures guidées
This lecture aims to introduce students to the idea of dimension reduction, highlighting basic strategies presented in the literature. In the first part, we address challenges and fundamental approaches in the study of the Dirichlet Laplacian on thin two-dimensional strips. Three models are considered—straight strips, curved strips, and strips with non-uniform width—illustrating how geometry influences the asymptotic behavior of eigenvalues as the width tends to zero. The corresponding effective operators obtained in this limit include the one-dimensional Laplacian and Schrödinger-type operators with geometric potentials.In the second part, we extend the discussion to strips embedded in three-dimensional space and equipped with mixed Dirichlet–Neumann boundary conditions. These results are inspired by and build upon previous works on the purely Dirichlet case, which serve as the main reference for our analysis. A comparison with the Dirichlet setting highlights both the analogies and the differences in the asymptotic behavior of eigenvalues as the strip becomes thin.
