Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps

Назаров, С. А.; Ruotsalainen, K.; Taskinen, Jari

doi:10.1080/00036810903479715

Cited by 25 publications

(33 citation statements)

References 14 publications

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“…A comparison of the present work with the paper [31] was already presented above. We just mention that the method of [31] is based on the max-min principle for eigenvalues. We finally also mention the paper [3], which contains an analysis of spectral bands much like in the present work, but in the more simple setting of the linear water wave equation.…”

Section: Introductionsupporting

confidence: 63%

Bands in the spectrum of a periodic elastic waveguide

Bakharev

Taskinen

2017

Z. Angew. Math. Phys.

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Section: Introductionsupporting

confidence: 63%

Bands in the spectrum of a periodic elastic waveguide

Bakharev

Taskinen

2017

Z. Angew. Math. Phys.

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“…We plug w(x) into (14) and pass to ε → 0. Using the same arguments as in the proof of Theorem 2.1 from [3] (but with account of (15)) we obtain:…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

Example of a periodic Neumann waveguide with a gap in its spectrum

Cardone

Khrabustovskyi

2017

EMS Series of Congress Reports

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In this note we investigate spectral properties of a periodic waveguide Ω ε (ε is a small parameter) obtained from a straight strip by attaching an array of ε-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator A ε = −ρ ε ∆Ωε , where ∆Ωε is the Neumann Laplacian in Ω ε , the weight ρ ε is equal to 1 everywhere except the union of the "rooms". We will prove that the spectrum of A ε has at least one gap as ε is small enough provided certain conditions on the weight ρ ε and the sizes of attached protuberances hold.

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“…Examples of periodic quantum, acoustic and elastic waveguides with gaps in their essential spectra have been presented in the literature, see [2,3,5,6,10,11,23,24,29], and [7,8,22,28], respectively. However, the approach of these papers, or the general approach of [21], cannot be applied to the piezoelectric waveguides of the present paper, because a direct weak formulation of the piezoelectricity problem does not correspond to a semi-bounded self-adjoint operator; the essential spectrum of such a problem would be non-physical, covering the whole plane C as shown in Sect.…”

Section: Motivationmentioning

confidence: 99%

“…For this reason, we do not treat the problem (5), (28), although all of our results can easily be adapted to this case, too.…”

Section: On the Physical Backgroundmentioning

confidence: 99%

“…In [28], standard local elliptic estimates were used to derive pointwise estimates for the eigenfunctions of the limit problem near points corresponding to O ± of this paper, and similar estimates will also be used here, see Lemma 3.1. However, the piezoelectricity case contains more terms to be estimated, in particular those related with the non-local operators, and following the scheme of [28] would also require pointwise estimates of the eigenfunctions of the η-dependent problem. These are not available, since the boundary of the domain ε is not smooth around the points O ± , but that problem can be circumvented by using the weighted Sobolev estimates of Lemma 3.2 and 4.1.…”

Section: Upper Bound For Spectral Bandsmentioning

confidence: 99%

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