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This article is cited in 14 scientific papers (total in 15 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.
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”, Sb. Math., 189:4 (1998), 503–526
Linking options:
https://www.mathnet.ru/eng/sm305https://doi.org/10.1070/sm1998v189n04ABEH000305 https://www.mathnet.ru/eng/sm/v189/i4/p25
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