Asymptotic expansion of eigenvalues of the neumann problem in a domain with a thin bridge

Назаров, С. А.; Polyakova, O. R.

doi:10.1007/bf00971127

Cited by 8 publications

(13 citation statements)

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“…Since the pioneering work [37] of E. Sanchez-Palencia, a lot of attention has been paid to mathematical analysis of vibrations of elastic bodies, with small parts wich are very heavy (e.g., pellets in an aspic or in a meat-jelly); see papers [4,9,12,25,30,33,39], as well as the monographs [34,38] in an incomplete list. Such problems are the best examples of the topping role of the boundary layer effect.…”

Section: Asymptotic Analysis Of Eigenvalues In Singularly Perturbed Dmentioning

confidence: 99%

“…Now, we evaluate the right-hand sides of the problems (29), (30). First, by the representation of the stiffness matrix…”

Section: Formal Construction Of Asymptoticsmentioning

confidence: 99%

“…General results of [7] (see also [[20]; §3.5, §6.1, §6.4]) show that there exists a unique decaying solution of problem (29), (30), which admits the expansion…”

Section: Formal Construction Of Asymptoticsmentioning

confidence: 99%

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On asymptotic analysis of spectral problems in elasticity

Sokołowski

2011

Lat. Am. j. solids struct.

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The three-dimensional spectral elasticity problem is studied in an anisotropic and inhomogeneous solid with small defects, i.e., inclusions, voids, and microcracks. Asymptotics of eigenfrequencies and the corresponding elastic eigenmodes are constructed and justified. New technicalities of the asymptotic analysis are related to variable coefficients of differential operators, vectorial setting of the problem, and usage of intrinsic integral characteristics of defects. The asymptotic formulae are developed in a form convenient for application in shape optimization and inverse problems. Keywords singular perturbations; spectral problem; asymptotics of eigenfunctions and eignevalues; elasticity boundary value problem.

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Section: Asymptotic Analysis Of Eigenvalues In Singularly Perturbed Dmentioning

confidence: 99%

“…Now, we evaluate the right-hand sides of the problems (29), (30). First, by the representation of the stiffness matrix…”

Section: Formal Construction Of Asymptoticsmentioning

confidence: 99%

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On asymptotic analysis of spectral problems in elasticity

Sokołowski

2011

Lat. Am. j. solids struct.

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“…Here we mention papers [21][22][23][24][25][26][27][28] where the rational dependence on |ln h| have been found in particular problems of mathematical physics and concomitant effects have been studied. We emphasize that asymptotic expansions [24,26,28] (see also [9,Chaps.…”

Section: Asymptotics Of Solutionsmentioning

confidence: 99%

“…We emphasize that asymptotic expansions [24,26,28] (see also [9,Chaps. 9,10]) in spectral problems get extremely intricate structure involving the holomorphic dependence on the parameter H in (2.10).…”

Section: Asymptotics Of Solutionsmentioning

confidence: 99%

On the rate of convergence for perforated plates with a small interior Dirichlet zone

Cardone

Назаров

Piatnitski

2010

Z. Angew. Math. Phys.

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The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neumann condition is assigned. In the second problem, Dirichlet condition is also imposed at the surface of one of the holes. Although in these two cases, the homogenized problem is the same, the asymptotic behavior of solutions is rather different. In particular, the presence of perturbation in the boundary condition in the second problem results in logarithmic rate of convergence, while for non-perturbed problem the rate of convergence is of power-law type. (2000). 35B25 · 35B27 · 35J25. Mathematics Subject Classification

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