We consider a family {Omega(epsilon)}epsilon>o of periodic domains in R-2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian -Delta(Omega)epsilon on Omega(epsilon). The waveguide Omega(epsilon) is a union of a thin straight strip of the width e and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period epsilon, along the strip upper boundary. We prove a (kind of) resolvent convergence of -Delta(Omega epsilon) to a certain operator on the line as epsilon -> 0. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of "passages" are appropriately scaled the first spectral gap of -Delta(Omega epsilon). is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory. (C) 2017 Elsevier Inc. All rights reserved.
Spectrum of a singularly perturbed periodic thin waveguide / Cardone, G; Khrabustovskyi, A.. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 454:2(2017), pp. 673-694. [10.1016/j.jmaa.2017.05.012]
Spectrum of a singularly perturbed periodic thin waveguide
Cardone G;
2017
Abstract
We consider a family {Omega(epsilon)}epsilon>o of periodic domains in R-2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian -Delta(Omega)epsilon on Omega(epsilon). The waveguide Omega(epsilon) is a union of a thin straight strip of the width e and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period epsilon, along the strip upper boundary. We prove a (kind of) resolvent convergence of -Delta(Omega epsilon) to a certain operator on the line as epsilon -> 0. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of "passages" are appropriately scaled the first spectral gap of -Delta(Omega epsilon). is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory. (C) 2017 Elsevier Inc. All rights reserved.File | Dimensione | Formato | |
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