Geodesic
On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 56-70 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A boundary-value problem of finding eigenvalues is considered for the negative Laplace operator in a disk with Neumann boundary condition on almost all circle except for a small arc of vanishing length, where the Dirichlet boundary condition is imposed. Complete asymptotic expansions with respect to a parameter (the length of the small arc) are constructed for an eigenvalue of this problem; the eigenvalue converges to a double eigenvalue of the Neumann problem.
Keywords: Laplace operator; singular perturbation; small parameter; eigenvalue; asymptotics. 
@article{TIMM_2015_21_1_a5,
     author = {R. R. Gadyl'shin and S. V. Repjevskij and E. A. Shishkina},
     title = {On an eigenvalue for the {Laplace} operator in a disk with {Dirichlet} boundary condition on a small part of the boundary in a critical case},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {56--70},
     year = {2015},
     volume = {21},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a5/}
}
TY  - JOUR
AU  - R. R. Gadyl'shin
AU  - S. V. Repjevskij
AU  - E. A. Shishkina
TI  - On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2015
SP  - 56
EP  - 70
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a5/
LA  - ru
ID  - TIMM_2015_21_1_a5
ER  - 
%0 Journal Article
%A R. R. Gadyl'shin
%A S. V. Repjevskij
%A E. A. Shishkina
%T On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case
%J Trudy Instituta matematiki i mehaniki
%D 2015
%P 56-70
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a5/
%G ru
%F TIMM_2015_21_1_a5
R. R. Gadyl'shin; S. V. Repjevskij; E. A. Shishkina. On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 56-70. http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a5/

[1] Gadylshin R.R., “Spektr ellipticheskikh kraevykh zadach pri singulyarnom vozmuschenii granichnykh uslovii”, Asimptot. svoistva reshenii differents. uravnenii, sb. nauch. st., Izd-vo BNTs UrO AN SSSR, 1988, 4–16 | MR

[2] Ilin A.M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp. | MR

[3] Gadylshin R.R., “Metod soglasovaniya asimptoticheskikh razlozhenii v singulyarno vozmuschennoi kraevoi zadache dlya operatora Laplasa”, Sovrem. matematika i ee prilozheniya, 5 (2003), 3–32 | Zbl

[4] Bikmetov A.R., Gadylshin R.R., “Vozmuschenie ellipticheskogo operatora uzkim potentsialom v $n$-mernoi oblasti”, Ufim. mat. zhurn, 4:2 (2012), 28–64

[5] Gadylshin R.R., “Rasscheplenie kratnogo sobstvennogo znacheniya v kraevoi zadache dlya membrany”, Sib. mat. zhurn, 34:3 (1993), 43–61 | MR | Zbl

[6] Gadylshin R.R., Shishkina E.A., “O neravenstvakh Fridrikhsa dlya kruga”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18:2 (2012), 48–61

[7] Chechkin G.A., “Usrednenie kraevykh zadach s singulyarnym vozmuscheniem granichnykh uslovii”, Mat. sb., 184:6 (1993), 99–150 | MR | Zbl

[8] Gadylshin R.R., “Asimptotiki sobstvennykh znachenii kraevoi zadachi s bystroostsilliruyuschimi granichnymi usloviyami”, Differents. uravneniya, 35:4 (1999), 540–551 | MR | Zbl

[9] Bakhvalov N.S., Panasenko G.P., Osrednenie protsessov v periodicheskikh sredakh. Matematicheskie zadachi mekhaniki kompozitsionnykh materialov, Nauka, M., 1984, 352 pp. | MR

[10] Oleinik O.A., Iosifyan G.A., Shamaev A.S., Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, Izd-vo Mosk. gos. un-ta, M., 1990, 311 pp. | MR

[11] Pyatnitskii A.L., Chechkin G.A., Shamaev A.S., Usrednenie. Metody i prilozheniya, Belaya seriya v matematike i fizike, Izd-vo “Tamara Rozhkovskaya”, Novosibirsk, 2004, 246 pp.

[12] Vladimirov V.S., Zharinov V.V., Uravneniya matematicheskoi fiziki, Fizmatlit, M., 2004, 400 pp.

[13] Ilin A.M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. 2. Oblast s malym otverstiem”, Mat. sb., 103(145):2(6) (1977), 265–284 | MR | Zbl

[14] Sobolev S.L., Vvedenie v teoriyu kubaturnykh formul, Nauka, M., 1974, 808 pp. | MR

[15] Gadylshin R.R., Asimptotika reshenii ellipticheskikh zadach s singulyarno vozmuschennymi granichnymi usloviya, diss. ... kand. fiz.-mat. nauk, Ufa, 1988, 133 pp.

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