Asymptotic expansion of eigenelements of the Laplace operator in a domain with a large number of ‘light’ concentrated masses sparsely situated on the boundary. Two-dimensional case

Chechkin, Gregory A.

doi:10.1090/s0077-1554-09-00177-0

Cited by 9 publications

(14 citation statements)

References 40 publications

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“…It was established that in the case α > 2, the eigenvalues have the following asymptotics:

λ_{n} (ϵ) = ϵ^{α - 2} Λ_{1} + ϵ^{α} (μ_{}^{n (1)} + θ_{ϵ}) + o (ϵ^{α}),

where Λ 1 and θ ϵ are related to the corresponding cell spectral problem and

\{μ_{n}^{(1)} : n \in double-struckN\}

are eigenvalues of the resulting spectral problem obtained after rescaling

λ_{n} (ϵ) = ϵ^{α - 2} (Λ_{ϵ}^{(1)} + ϵ^{2} μ_{n} (ϵ)), u_{n} (ϵ, x) = ϵ Z_{ϵ} (\frac{x}{ϵ}) U_{n} (ϵ, x)

of the initial spectral problem. Comparing the main results with the ones in the papers mentioned earlier, we see that both the eigenvalues in and eigenvalues are infinitely small, of order

O (ϵ^{α - 2})

, but in , splitting of the asymptotics for the eigenvalues occurs only in the second term. The first leading terms of the asymptotics both for the eigenvalue and eigenfunctions were constructed and justified in for different values of the parameter α : α < 2, α = 2, α > 2.…”

Section: Conclusion and Remarkssupporting

confidence: 64%

“…Results for three‐dimensional (3D) case are quite different because the asymptotics of the corresponding boundary‐layer solutions are different (see ). It should be noted here that the geometry of problems in and in this paper is significantly different from the geometry of , where only perturbation of the density on sparse inclusions is present.…”

Section: Conclusion and Remarksmentioning

confidence: 96%

“…Spectral problems with concentrated masses near the boundary were considered in . In , the authors considered vibrations of a domain containing many small regions of high density of order

O (ϵ^{- α})

, periodically situated along the boundary.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

“…In , the author considered the analog problem as in and constructed asymptotic expansions for eigenvalues and eigenfunctions in the case α < 2;

η MathClass-rel= scriptO MathClass-open(ϵ MathClass-close)

in , and η ln ϵ → 0 as ϵ → 0 in .…”

Section: Conclusion and Remarksmentioning

confidence: 99%

“…Using the approach of the present paper and papers , it is possible also to study the problems mentioned in . Moreover, we can consider spectral problems with more general settings, for instance, in domains with adjoint small concentrated masses near the boundary (see the first picture of Figure , concentrated masses are in black color).…”

Section: Conclusion and Remarksmentioning

confidence: 99%

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Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

Chechkin

Mel'nyk

2013

Math Methods in App Sciences

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A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number 5N=O(ϵ−1) of ϵ‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order scriptOMathClass-open(ϵMathClass-close). The density of the junction is of order O(ϵ−α) on the rods from the second class and scriptOMathClass-open(1MathClass-close) outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ϵ → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ϵ → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.

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“…It was established that in the case α > 2, the eigenvalues have the following asymptotics:

λ_{n} (ϵ) = ϵ^{α - 2} Λ_{1} + ϵ^{α} (μ_{}^{n (1)} + θ_{ϵ}) + o (ϵ^{α}),

where Λ 1 and θ ϵ are related to the corresponding cell spectral problem and

\{μ_{n}^{(1)} : n \in double-struckN\}

are eigenvalues of the resulting spectral problem obtained after rescaling

λ_{n} (ϵ) = ϵ^{α - 2} (Λ_{ϵ}^{(1)} + ϵ^{2} μ_{n} (ϵ)), u_{n} (ϵ, x) = ϵ Z_{ϵ} (\frac{x}{ϵ}) U_{n} (ϵ, x)

of the initial spectral problem. Comparing the main results with the ones in the papers mentioned earlier, we see that both the eigenvalues in and eigenvalues are infinitely small, of order

O (ϵ^{α - 2})

Section: Conclusion and Remarkssupporting

confidence: 64%

Section: Conclusion and Remarksmentioning

confidence: 96%

“…Spectral problems with concentrated masses near the boundary were considered in . In , the authors considered vibrations of a domain containing many small regions of high density of order

O (ϵ^{- α})

, periodically situated along the boundary.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

“…In , the author considered the analog problem as in and constructed asymptotic expansions for eigenvalues and eigenfunctions in the case α < 2;

η MathClass-rel= scriptO MathClass-open(ϵ MathClass-close)

in , and η ln ϵ → 0 as ϵ → 0 in .…”

Section: Conclusion and Remarksmentioning

confidence: 99%

Section: Conclusion and Remarksmentioning

confidence: 99%

See 3 more Smart Citations

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

Chechkin

Mel'nyk

2013

Math Methods in App Sciences

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Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

Bakharev

Pérez

2017

Mathematische Nachrichten

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We consider a spectral problem for the Laplace operator in a periodic waveguide Π⊂R3 perturbed by a family of “heavy concentrated masses”; namely, Π contains small regions false{ωjεfalse}j∈double-struckZ of high density, which are periodically distributed along the z axis. Each domain ωjε⊂Π has a diameter O(ε) and the density takes the value ε−m in ωjε and 1 outside; m and ε are positive parameters, m>2, ε≪1. Considering a Dirichlet boundary condition, we study the band‐gap structure of the essential spectrum of the corresponding operator as ε→0. We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.

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Perturbation by Slender Potential of Operators Associated with Sectorial Forms

2014

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Asymptotic expansion of eigenelements of the Laplace operator in a domain with a large number of ‘light’ concentrated masses sparsely situated on the boundary. Two-dimensional case

Cited by 9 publications

References 40 publications

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

Perturbation by Slender Potential of Operators Associated with Sectorial Forms

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