Asymptotic analysis of boundary value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions

Mel’nik, T. A.; Popov, A. V.

doi:10.1070/sm2012v203n08abeh004259

Cited by 10 publications

(7 citation statements)

References 9 publications

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“…More general geometries were treated in [3], where thin strip is perturbed by attaching small protuberances, ε α -periodically along it boundary; the protuberances are obtained from a fixed bounded domain by ε α -rescaling in x 1 direction and ε-rescaling in x 2 direction. In [22,23], besides an oscillating external boundary, additional internal holes are allowed.…”

Section: Introductionmentioning

confidence: 99%

Spectrum of a singularly perturbed periodic thin waveguide

Cardone

Khrabustovskyi

2017

Journal of Mathematical Analysis and Applications

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We consider a family {Ω ε } ε>0 of periodic domains in R 2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian −∆ Ω ε on Ω ε . The waveguide Ω ε is a union of a thin straight strip of the width ε and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period ε, along the strip upper boundary. For ε → 0 we prove a (kind of) resolvent convergence of −∆ Ω ε to a certain ordinary differential operator. 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 −∆ Ω ε is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.

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Section: Introductionmentioning

confidence: 99%

Spectrum of a singularly perturbed periodic thin waveguide

Cardone

Khrabustovskyi

2017

Journal of Mathematical Analysis and Applications

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“…Remark 2.2. By using the recursive procedure for the boundary-value problem (7), one can easily show that the functions u (i) 2p+1 are identically equal to zero for odd k = 2p + 1, p ∈ N.…”

Section: Formal Asymptotic Expansion 21 Regular Part Of the Asymptoticsmentioning

confidence: 99%

“…5), or graph-junctions of thin perforated domains with rapidly varying thickness. In the last case, it is necessary to add series with rapidly oscillating coefficients to the regular part of the asymptotics (see [7]). Figure 5: A graph-junction of thin domains with a local joint…”

Section: Justificationmentioning

confidence: 99%

Asymptotic expansion for the solution to a boundary-value problem in a thin cascade domain with a local joint

Klevtsovskiy

Mel'nyk

2016

ASY

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A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain Ω ε coinciding with two thin rectangles connected through a joint of diameter O(ε) . A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter ε → 0. Energetic and uniform pointwise estimates for the difference between the solution of the starting problem (ε > 0) and the solution of the corresponding limit problem (ε = 0) are proved, from which the influence of the geometric irregularity of the joint is observed.

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“…We first mention the pioneering works [10,11], as well the subsequent papers [12,13,14], where the authors investigate the asymptotic behavior of dynamical systems given by a class of semilinear parabolic equations in thin domains of R n , n ≥ 2. We also cite [15,16] where the p-Laplacian problem in thin regions is considered, and [17], which studies a linear elliptic problem in perforated thin domains with rapidly varying thickness. In [18] the authors consider nonlinear monotone problems in a multidomain with a highly oscillating boundary.…”

Section: Introductionmentioning

confidence: 99%

Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

Arrieta

Nogueira

Pereira

2019

Computers & Mathematics with Applications

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In this work we study the behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary. Our main result is concerned with the upper and lower semicontinuity of the set of solutions. We show that the solutions of our perturbed equation can be approximated with ones of a one-dimensional equation, which also captures the effects of all relevant physical processes that take place in the original problem.2010 Mathematics Subject Classification. 34B15, 35J75, 35J91.

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Asymptotic analysis of boundary value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions

Cited by 10 publications

References 9 publications

Spectrum of a singularly perturbed periodic thin waveguide

Spectrum of a singularly perturbed periodic thin waveguide

Asymptotic expansion for the solution to a boundary-value problem in a thin cascade domain with a local joint

Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

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