What question did this study set out to answer?

This research aims to analyze the spectral properties of the Laplacian in 3D rods and how they relate to 1D models.

January 24, 2026

Asymptotics for the Spectrum of the Laplacian in Thin Bars with Varying Cross Sections

Key Points

This research aims to analyze the spectral properties of the Laplacian in 3D rods and how they relate to 1D models.
Analyzed the Laplace operator in 3D structures with varying diameters.
Considered Dirichlet and Neumann boundary conditions on different surfaces.
Explored low frequency eigenfunctions in Sobolev spaces.
Demonstrated that the spectrum converges to a 1D spectral model as the diameter approaches zero.
Maintained the multiplicity of spectral values during convergence.
Showed that geometric variations significantly affect spectral behavior.

Abstract

ABSTRACT We consider spectral problems for the 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 toward that of a 1D spectral model with Dirichlet (Neumann, respectively) boundary conditions. This 1D model may arise in diffusion or vibration 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.

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