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 
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/
@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},
     year = {1998},
     volume = {189},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_4_a1/}
}
TY  - JOUR
AU  - R. R. Gadyl'shin
AU  - A. M. Il'in
TI  - Asymptotic behaviour of the eigenvalues of the Dirichlet problem in a domain with a narrow slit
JO  - Sbornik. Mathematics
PY  - 1998
SP  - 503
EP  - 526
VL  - 189
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1998_189_4_a1/
LA  - en
ID  - SM_1998_189_4_a1
ER  - 
%0 Journal Article
%A R. R. Gadyl'shin
%A A. M. Il'in
%T Asymptotic behaviour of the eigenvalues of the Dirichlet problem in a domain with a narrow slit
%J Sbornik. Mathematics
%D 1998
%P 503-526
%V 189
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1998_189_4_a1/
%G en
%F SM_1998_189_4_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Samarskii A. A., “O vliyanii zakrepleniya na sobstvennye chastoty zamknutykh ob'emov”, Dokl. AN SSSR, 63 (1948), 631–634 | MR | Zbl

[2] Ozawa S., “Singular variation of domains and eigenvalues of the Laplacian”, Duke Math. J., 48 (1981), 767–778 | DOI | MR | Zbl

[3] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Asimptoticheskie razlozheniya sobstvennykh chisel kraevykh zadach dlya operatora Laplasa v oblastyakh s malymi otverstiyami”, Izv. AN SSSR. Ser. matem., 48:2 (1984), 347–371 | MR

[4] Gadylshin R. R., “Asimptotika sobstvennogo znacheniya singulyarno vozmuschennoi ellipticheskoi zadachi s malym parametrom v granichnom uslovii”, Differents. uravneniya, 22:4 (1986), 640–652 | MR | Zbl

[5] Nazarov S. A., Asimptoticheskie razlozheniya sobstvennykh chisel, Izd-vo LGU, L., 1987

[6] Gadylshin R. R., “Rasscheplenie kratnogo sobstvennogo znacheniya zadachi Dirikhle dlya operatora Laplasa pri singulyarnom vozmuschenii granichnogo usloviya”, Matem. zametki, 52:4 (1992), 42–55 | MR | Zbl

[7] Jimbo S., Morita Y., “Remarks on the behavior of certain eigenvalues in the singularly perturbed domain with several thin channels”, Comm. Partial Differential Equations, 17 (1992), 523–552 | MR | Zbl

[8] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993 | MR | Zbl

[9] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. 1: Dvumernyi sluchai”, Matem. sb., 99:4 (1976), 514–537 | MR | Zbl

[10] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[11] Van-Daik M., Metody vozmuschenii v mekhanike zhidkosti, Mir, M., 1967

[12] Naife A. Kh., Metody vozmuschenii, Mir, M., 1986

[13] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[14] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi i uglovymi tochkami”, Tr. MMO, 16, URSS, M., 1967, 209–292 | MR | Zbl