Geodesic
Asymptotics of the Eigenelements of the Laplace Operator when the Boundary-Condition Type Changes on a Narrow Flattened Strip
Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 236-252.

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In this paper, the method of matching asymptotic expansions is used to construct an asymptotics (in a small parameter) of the eigenvalues and eigenfunctions of the Laplace operator in a domain when the boundary-condition type changes on a narrow flattened strip, provided that on the narrow strip of the boundary a Neumann condition is given and on the remaining part of the boundary a Dirichlet condition is given. The width of the strip is taken as the small parameter. 
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M. Yu. Planida. Asymptotics of the Eigenelements of the Laplace Operator when the Boundary-Condition Type Changes on a Narrow Flattened Strip. Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 236-252. http://geodesic.mathdoc.fr/item/MZM_2004_75_2_a6/

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

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

[3] Dnestrovskii Yu. N., “Ob izmenenii sobstvennykh chisel pri izmenenii granitsy oblastei”, Vestn. MGU, 1964, no. 9, 61–74 | MR

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

[5] Mazya V. G., Nazarov S. P., 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

[6] Gadylshin R. R., Ilin A. M., “Asimptotika sobstvennykh znachenii zadachi Dirikhle v oblasti s uzkoi schelyu”, Matem. sb., 189:4 (1998), 25–48 | MR

[7] Ilin A. M., Gadylshin R. R., “O granitse primenimosti regulyarnoi teorii vozmuschenii dlya membrany s uzkoi schelyu”, Dokl. RAN, 380:2 (2001), 172–175 | MR

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

[9] 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

[10] Gadylshin R. R., “O vozmuschenii spektra laplasiana pri smene tipa granichnogo usloviya na maloi chasti granitsy”, ZhVMiMF, 1996, no. 7, 77–88 | MR

[11] Argatov I. I., Nazarov S. A., “Asimptoticheskoe reshenie zadachi Sinorini s prepyatstviem na tonkom prodolgovatom mnozhestve”, Matem. sb., 187:10 (1996), 3–32 | MR | Zbl

[12] Planida M. Yu., “O skhodimosti reshenii singulyarno vozmuschennykh kraevykh zadach dlya laplasiana”, Matem. zametki, 71:6 (2002), 867–877 | MR | Zbl

[13] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1977 | MR | Zbl

[14] Borisov D. I., “O singulyarno vozmuschennoi kraevoi zadache dlya laplasiana v tsilindre”, Differents. uravneniya, 38:8 (2002), 1071–1078 | MR | Zbl

[15] Vishik M. I., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, UMN, 12:5 (1957), 3–122 | MR | Zbl