Asymptotics for the spectrum of the Laplacian in thin bars with varying cross sections

TL;DR

This paper analyzes how the spectrum of the Laplacian in thin 3D rod structures converges to a 1D model as the cross section shrinks, providing insights into spectral behavior in physical models with varying geometries.

Contribution

It establishes the spectral convergence of the Laplacian in thin rods with varying cross sections to a 1D model, including eigenfunction behavior and boundary condition effects.

Findings

Spectrum converges to 1D model as cross section shrinks

Eigenvalues and multiplicities are preserved in the limit

Approach to eigenfunctions in Sobolev spaces is characterized

Abstract

We consider spectral problems for Laplace operator in 3D rod structures with a small cross section of diameter $O(\varepsilon)$, $\varepsilon$ being a positive parameter. The boundary conditions are Dirichlet (Neumann, respectively) on the bases of this structure and Neumann on the lateral boundary. As $\varepsilon\to 0$, we show the convergence of the spectrum with conservation of the multiplicity towards that of a 1D spectral model with Dirichlet (Neumann, respectively) boundary conditions. This 1D model may arise in diffusion or vibrations models of nonhomogeneous media with different physical characteristics and it takes into account the geometry of the 3D domain. We deal with the low frequencies and the approach to eigenfunctions in the suitable Sobolev spaces is also outlined.