Spectrum of the Dirichlet Laplacian in a thin cubic lattice

¹ Inria, Ensta Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91762 Palaiseau, France
² Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoi pr., 61, St. Petersburg 199178, Russia

^* Corresponding author: lucas.chesnel@inria.fr

Received: 14 January 2023
Accepted: 2 October 2023

Abstract

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width ε ≪ 1) which have a square cross section. This spectrum coincides with the union of segments which all go to +∞ as ε tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length O(e^−δ/ε), δ > 0, while the length of the next spectral segments is O(ε). To establish these results, we need to study in detail the properties of the Dirichlet Laplacian A^Ω in the geometry Ω obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max–min arguments as well as a well-chosen Poincaré–Friedrichs inequality, we prove that A^Ω has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for A^Ω, that is no non trivial bounded solution at the threshold frequency for A^Ω. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.