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

Pablo Benavent-Ocejo, Delfina Gómez, Maria-Eugenia Pérez-Martínez

TL;DR

The paper develops a rigorous dimension-reduction framework for the Laplacian in thin rod-like domains with varying cross sections, proving that the 3D spectrum converges to a weighted 1D eigenproblem on the longitudinal interval as the cross-section shrinks. Using a stretching-variable change of variables and a spectral perturbation lemma, it demonstrates multiplicity-preserving convergence for both mixed Dirichlet/Neumann and Neumann boundary conditions, with the cross-section geometry encoded via the weight |D_{x_1}|. The results yield explicit limit problems and Sobolev-space convergence for low-frequency eigenfunctions, providing a rigorous basis for diffusion and vibration models in nonhomogeneous media. The methodology combines variational formulations, dimension reduction, and operator convergence, and highlights remaining questions for high-frequency behavior and numerical approximation as the cross-section becomes extremely small.

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.

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

TL;DR

The paper develops a rigorous dimension-reduction framework for the Laplacian in thin rod-like domains with varying cross sections, proving that the 3D spectrum converges to a weighted 1D eigenproblem on the longitudinal interval as the cross-section shrinks. Using a stretching-variable change of variables and a spectral perturbation lemma, it demonstrates multiplicity-preserving convergence for both mixed Dirichlet/Neumann and Neumann boundary conditions, with the cross-section geometry encoded via the weight |D_{x_1}|. The results yield explicit limit problems and Sobolev-space convergence for low-frequency eigenfunctions, providing a rigorous basis for diffusion and vibration models in nonhomogeneous media. The methodology combines variational formulations, dimension reduction, and operator convergence, and highlights remaining questions for high-frequency behavior and numerical approximation as the cross-section becomes extremely small.

Abstract

We consider spectral problems for Laplace operator in 3D rod structures with a small cross section of diameter , being a positive parameter. The boundary conditions are Dirichlet (Neumann, respectively) on the bases of this structure and Neumann on the lateral boundary. As , 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.
Paper Structure (6 sections, 3 theorems, 65 equations, 5 figures)

This paper contains 6 sections, 3 theorems, 65 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Preliminary results
  4. The convergence for the mixed boundary value problem
  5. The convergence for the Neumann problem
  6. Concluding Remarks

Key Result

Lemma 3.1

Let $\mathcal{H}_\varepsilon$ and $\mathcal{H}_0$ be two separable Hilbert spaces with the scalar products $\langle\cdot, \cdot\rangle_{\mathcal{H}_\varepsilon}$ and $\langle\cdot, \cdot\rangle_{\mathcal{H}_0}$ respectively. Let $\mathcal{A}_\varepsilon\in \mathcal{L}(\mathcal{H}_\varepsilon)$ and $ Let $\{\mu_\varepsilon^i\}_{i=1}^\infty$ ($\{\mu_0^i\}_{i=1}^\infty$, respectively) be the sequence

Figures (5)

  • Figure 1: Geometrical configuration of a domain $G_\varepsilon$.
  • Figure 2: Polyhedral domains which are the union of two cuboids of different cross sections (of size $O(\varepsilon)$ in both transverse directions) or union of cuboids with different heights $O(\varepsilon)$.
  • Figure 3: Dome-like top or wavy-like top domain of size $O(\varepsilon)$ in both transverse directions and trumpet-like domain of diameter $O(\varepsilon)$.
  • Figure 4: Graphs based on numerical approximations of eigenfunctions corresponding to the first positive eigenvalue of the mixed boundary value problem (above) and the Neumann one (below), for two domains $G_\varepsilon$ like those in Figures \ref{['ej_unionprismas']}-\ref{['figura_boveda']}.
  • Figure 5: Graphs based on numerical approximations of some eigenfunctions of the mixed boundary value problem (above) and the Neumann one (below), for two domains $G_\varepsilon$ like those in Figures \ref{['ej_unionprismas']}-\ref{['figura_boveda']}.

Theorems & Definitions (3)

  • Lemma 3.1
  • Theorem 4.1
  • Theorem 5.1