Longitudinal Oscillations for Eigenfunctions in Rod Like Structures

Pablo Benavent-Ocejo (), Delfina Gómez () and María-Eugenia Pérez-Martínez ()
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Pablo Benavent-Ocejo: Universidad de Cantabria
Delfina Gómez: Universidad de Cantabria
María-Eugenia Pérez-Martínez: Universidad de Cantabria

Chapter Chapter 1 in Integral Methods in Science and Engineering, 2026, pp 1-21 from Springer

Abstract: Abstract We consider the spectrum of the Laplace operator on 3D rod structures, with a small cross section depending on a small parameter ε $$\varepsilon $$ . The boundary conditions are of Dirichlet type on the bases 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 ε → 0 $$\varepsilon \to 0$$ 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.

Date: 2026
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DOI: 10.1007/978-3-032-04458-7_1

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