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Mat. Sb., 1998, Volume 189, Number 4, Pages 25–48 (Mi msb305)  

This article is cited in 11 scientific papers (total in 12 papers)

Asymptotic behaviour of the eigenvalues of the Dirichlet problem in a domain with a narrow slit

R. R. Gadyl'shina, A. M. Il'inb

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: The Dirichlet problem in a two-dimensional domain with a narrow slit is studied. The width of the slit is a small parameter. The complete asymptotic expansion for the eigenvalue of the perturbed problem converging to a simple eigenvalue of the limiting problem is constructed by means of the method of matched asymptotic expansions. It is shown that the regular perturbation theory can formally be applied in a natural way up to terms of order $\varepsilon ^2$. However, the result obtained in that way is false. The correct result can be obtained only by means of an inner asymptotic expansion.


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English version:
Sbornik: Mathematics, 1998, 189:4, 503–526

Bibliographic databases:

UDC: 517.956
MSC: Primary 35C20; Secondary 35J25
Received: 26.05.1997

Citation: R. R. Gadyl'shin, A. M. Il'in, “Asymptotic behaviour of the eigenvalues of the Dirichlet problem in a domain with a narrow slit”, Mat. Sb., 189:4 (1998), 25–48; Sb. Math., 189:4 (1998), 503–526

Citation in format AMSBIB
\by R.~R.~Gadyl'shin, A.~M.~Il'in
\paper Asymptotic behaviour of the~eigenvalues of the~Dirichlet problem in a~domain with a~narrow slit
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 4
\pages 25--48
\jour Sb. Math.
\yr 1998
\vol 189
\issue 4
\pages 503--526

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    This publication is cited in the following articles:
    1. Il'in, AM, “On the limits of applicability of the regular perturbation theory to a membrane with a narrow aperture”, Doklady Mathematics, 64:2 (2001), 270  mathscinet  zmath  isi
    2. V. M. Babich, L. A. Kalyakin, M. D. Ramazanov, N. Kh. Rozov, “Arlen Mikhailovich Il'in (on the occasion of the 70th anniversary)”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S1–S7  mathnet  mathscinet  zmath  elib
    3. Ammari H., Triki F., “Resonances for microstrip transmission lines”, SIAM J. Appl. Math., 64:2 (2003), 601–636  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    4. Ammari H., Khelifi A., “Electromagnetic scattering by small dielectric inhomogeneities”, J. Math. Pures Appl. (9), 82:7 (2003), 749–842  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    5. Ammari, H, “Splitting of resonant and scattering frequencies under shape deformation”, Journal of Differential Equations, 202:2 (2004), 231  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    6. M. Yu. Planida, “Asymptotics of the Eigenelements of the Laplace Operator when the Boundary-Condition Type Changes on a Narrow Flattened Strip”, Math. Notes, 75:2 (2004), 213–228  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. M. Yu. Planida, “Asymptotics of the eigenelements of the Laplacian with singular perturbations of boundary conditions on narrow and thin sets”, Sb. Math., 196:5 (2005), 703–741  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Khelifi, A, “Asymptotic property and convergence estimation for the eigenelements of the Laplace operator”, Applicable Analysis, 86:10 (2007), 1249  crossref  mathscinet  zmath  isi
    9. Ammari H., Kang H., Lim M., Zribi H., “Layer Potential Techniques in Spectral Analysis. Part I: Complete Asymptotic Expansions for Eigenvalues of the Laplacian in Domains with Small Inclusions”, Transactions of the American Mathematical Society, 362:6 (2010), 2901–2922  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    10. Gadyl'shin R.R., Il'in A.M., “On the spectrum of the Neumann problem for Laplace equation in a domain with a narrow slit”, Asymptotic Analysis, 67:3–4 (2010), 167–189  crossref  mathscinet  zmath  isi  elib
    11. “Arlen Mikhailovich Ilin (k vosmidesyatiletiyu so dnya rozhdeniya)”, Ufimsk. matem. zhurn., 4:2 (2012), 3–12  mathnet  mathscinet
    12. A. A. Ershov, “Asimptotika resheniya vtoroi kraevoi zadachi dlya uravneniya Laplasa vne maloi okrestnosti otrezka”, Tr. IMM UrO RAN, 21, no. 1, 2015, 81–96  mathnet  mathscinet  elib
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