Geodesic
Asymptotic behaviour of the~eigenvalues of the~Dirichlet problem in a~domain with a~narrow slit
Sbornik. Mathematics, Tome 189 (1998) no. 4, pp. 503-526

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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. 
@article{SM_1998_189_4_a1,
     author = {R. R. Gadyl'shin and A. M. Il'in},
     title = {Asymptotic behaviour of the~eigenvalues of {the~Dirichlet} problem in a~domain with a~narrow slit},
     journal = {Sbornik. Mathematics},
     pages = {503--526},
     publisher = {mathdoc},
     volume = {189},
     number = {4},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_4_a1/}
}
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R. R. Gadyl'shin; A. M. Il'in. Asymptotic behaviour of the~eigenvalues of the~Dirichlet problem in a~domain with a~narrow slit. Sbornik. Mathematics, Tome 189 (1998) no. 4, pp. 503-526. http://geodesic.mathdoc.fr/item/SM_1998_189_4_a1/