Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses

Math Methods in App Sciences

2013

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.

P_{ϵ} MathClass-punc: H_{ϵ} MathClass-rel\mapsto H^{1} MathClass-open(Ω MathClass-punc, Γ_{1} MathClass-close)

∥ P_{ϵ} v_{n} (ϵ, \cdot) ∥_{H^{1} (Ω, Γ_{1})} \leq C_{n} ∥ v_{n} (ϵ, \cdot) ∥_{{scriptH}_{ϵ}} \leq C'_{n},

where Ω is the interior of the union

{\overset{true¯}{Ω}}_{0} MathClass-bin\cup \overset{D_{2}}{falsemml-overline}

.…”

Section: Justification Of the Asymptoticsmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 83%

“…We formulate the next theorem under the assumption that all

λ_{α - 1}^{(n + i)}

, i = 1, … , r , are different and

λ_{α - 1}^{(n + 1)} < λ_{α - 1}^{(n + 2)} < \dots < λ_{α - 1}^{(n + r)} .

In general case, the formulation is similar as in Theorems 5.4 and 5.6 from our paper . Theorem Let inequalities are satisfied.…”

Section: Justification Of the Asymptoticsmentioning

confidence: 95%

Section: Conclusion and Remarksmentioning

confidence: 96%

Section: Construction Of the Leading Terms Of The Asymptotics In The mentioning

confidence: 99%

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

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Asymptotic analysis of spectral problems in thick junctions with the branched fractal structure

Math Methods in App Sciences

2022

Self Cite

Funding informationThe Grant of the Ministry of Education and Science of Ukraine A spectral problem is considered in a domain Ω 𝜀 that is the union of a domain Ω 0 and a lot of thin trees situated 𝜀-periodically along some manifold on the boundary of Ω 0 . The trees have finite number of branching levels. The perturbed Robin boundary condition 𝜕 𝜈 u 𝜀 + 𝜀 𝛼 i k i,m u 𝜀 = 0 is given on the ith branching layer; {𝛼 i } are real parameters. The asymptotic analysis of this problem is made as 𝜀 → 0, that is, when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the Hausdorff convergence of the spectrum to the spectrum of the corresponding nonstandard homogenized spectral problem is proved, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.

Homogenization of a parabolic signorini boundary value problem in a thick plane junction

Nakvasiuk

2012

J Math Sci

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω ε which is the union of a domain Ω 0 and a large number of ε−periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as ε → 0) in differential inequalities in the region that is filled up by the thin rods in the limit passage. Bibliography: 31 titles. Illustrations: 1 figure.