Longitudinal oscillations for eigenfunctions in rod like structures

TL;DR

This paper investigates the asymptotic behavior of eigenvalues and eigenfunctions of the Laplace operator on thin 3D rod structures, revealing how geometry and boundary conditions influence spectral properties as the cross section shrinks.

Contribution

It provides a detailed asymptotic analysis of eigenfunctions and eigenvalues in rod-like structures, incorporating geometry effects and boundary conditions, with explicit and numerical results.

Findings

Eigenvalues approach a 1D model as cross section shrinks

Oscillations in transverse directions require high-frequency analysis

Boundary conditions significantly affect spectral asymptotics

Abstract

We consider the spectrum of the Laplace operator on 3D rod structures, with a small cross section depending on a small parameter $\varepsilon$. The boundary conditions are of Dirichlet type on the basis of this structure and Neumann on the lateral boundary. We focus on the low frequencies. We study the asymptotic behavior of the eigenvalues and associated eigenfunctions, which are approached as $\varepsilon\to 0$ by those of a 1D model with Dirichlet boundary conditions, but which takes into account the geometry of the domain. Explicit and numerical computations enlighten the interest of this study, when the parameter becomes smaller. At the same time they show that in order to capture oscillations in the transverse direction we need to deal with the high frequencies. For prism like domains, we show the different asymptotic behavior of the spectrum depending on the boundary conditions.