A gap in the spectrum of the Neumann–Laplacian on a periodic waveguide

Abstract

We study a spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder which contains a periodic arrangement of inclusions. On the boundary of the waveguide, we consider both Neumann and Dirichlet conditions. We prove that provided the diameter of the inclusion is small enough the spectrum of Laplace operator contains band gaps, i.e. there are frequencies that do not propagate through the waveguide. The existence of the band gaps is verified using the asymptotic analysis of elliptic operators.

Keywords:

• Helmholtz equation
• periodic waveguide
• spectral gaps
• singularly perturbed domains

AMS Subject Classifications::

• 35P05
• 34E05
• 34E10
• 76M45

Acknowledgements

The first author was supported by the Chebyshev Laboratory under RF government grant Nos. 11.G34.31.0026 and 6.38.64.2012. The second author was supported by Russian Foundation on Basic Research grant 12-01-00348.

Notes

Note

1. This does not hold in the two-dimensional case, where the existence of a gap is still fully an open question.