Geodesic
Singularly perturbed Dirichlet boundary value problem for a stationary system in the linear elasticity theory
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2008), pp. 7-16.

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

We consider a singularly perturbed Dirichlet boundary value problem for an elliptic operator of the linear elasticity theory in a bounded domain with a small cavity. The main result is the proof of the theorem about the convergence of eigenelements of the perturbed boundary value problem to eigenelements of the corresponding limit boundary value problem, when the parameter $\varepsilon$ which defines the diameter of the small cavity tends to zero.
Keywords: operator, boundary value problem
Mots-clés : singular perturbation, eigenelements. 
@article{IVM_2008_12_a1,
     author = {D. B. Davletov},
     title = {Singularly perturbed {Dirichlet} boundary value problem for a stationary system in the linear elasticity theory},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {7--16},
     publisher = {mathdoc},
     number = {12},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2008_12_a1/}
}
TY  - JOUR
AU  - D. B. Davletov
TI  - Singularly perturbed Dirichlet boundary value problem for a stationary system in the linear elasticity theory
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2008
SP  - 7
EP  - 16
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2008_12_a1/
LA  - ru
ID  - IVM_2008_12_a1
ER  - 
%0 Journal Article
%A D. B. Davletov
%T Singularly perturbed Dirichlet boundary value problem for a stationary system in the linear elasticity theory
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2008
%P 7-16
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2008_12_a1/
%G ru
%F IVM_2008_12_a1
D. B. Davletov. Singularly perturbed Dirichlet boundary value problem for a stationary system in the linear elasticity theory. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2008), pp. 7-16. http://geodesic.mathdoc.fr/item/IVM_2008_12_a1/

[1] Samarskii A. A., “O vliyanii zakrepleniya na sobstvennye chastoty zamknutykh ob'emov”, DAN SSSR, 63:6 (1948), 631–634 | MR

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

[3] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s tonkoi schelyu. 2: Oblast s malym otverstiem”, Matem. sb., 103:2 (1977), 265–284 | MR

[4] Ilin A. M., “Issledovanie asimptotiki resheniya ellipticheskoi kraevoi zadachi s malym otverstiem”, Tr. seminara im. I. G. Petrovskogo, no. 6, MGU, M., 1981, 57–82

[5] 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, 48:2 (1984), 347–371 | MR

[6] Kamotskii I. V., Nazarov S. A., “Spektralnye zadachi v singulyarno vozmuschennykh oblastyakh i samosopryazhennye rasshireniya differentsialnykh operatorov”, Tr. Sankt-Peterburgskogo matem. ob-va, 6, 1998, 151–212 | MR

[7] Oleinik O. A., Iosifyan G. A., Shamaev A. S., Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, MGU, M., 1990, 311 pp.

[8] Mazya V. G., Prostranstva Soboleva, LGU, L., 1985, 416 pp.

[9] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972, 740 pp. | MR | Zbl

[10] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983, 424 pp.

[11] Gadylshin R. R., “Spektr ellipticheskikh kraevykh zadach pri singulyarnom vozmuschenii granichnykh uslovii”, Asimptoticheskie svoistva reshenii differentsialnykh uravnenii, sb. nauchn. tr., BNTs UrO AN SSSR, Ufa, 1988, 3–15

[12] Gadylshin R. R., “O sobstvennykh chastotakh tel s tonkimi otrostkami. I: Skhodimost i otsenki”, Matem. zametki, 54:6 (1993), 10–21 | MR