Geodesic
Spectral Asymptotics for a Steady-State Heat Conduction Problem in a Perforated Domain
Matematičeskie zametki, Tome 69 (2001) no. 4, pp. 600-612 
S. E. Pastukhova. Spectral Asymptotics for a Steady-State Heat Conduction Problem in a Perforated Domain. Matematičeskie zametki, Tome 69 (2001) no. 4, pp. 600-612. http://geodesic.mathdoc.fr/item/MZM_2001_69_4_a7/
@article{MZM_2001_69_4_a7,
     author = {S. E. Pastukhova},
     title = {Spectral {Asymptotics} for a {Steady-State} {Heat} {Conduction} {Problem} in a {Perforated} {Domain}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {600--612},
     year = {2001},
     volume = {69},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_69_4_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we study the eigenvalues and eigenfunctions of a boundary-value problem for an elliptic equation of second order with oscillatory coefficients in a periodically perforated domain when the boundary condition on the external boundary is of the first type and on the boundary of “holes” of the third type, for the case in which the linear dimension $\varepsilon$ of the perforation period tends to zero. It is proved that these eigenvalues and eigenfunctions can be determined approximately via the eigenvalues and eigenfunctions of an essentially simpler Dirichlet problem for an elliptic equation with constant coefficients in a domain without holes. Estimates of errors in these approximations are given.

[1] Oleinik O. A., Iosifyan G. A., Shamaev A. S., Matematicheskie zadachi teorii silno neodnorodnykh uprugikh tel, Izd-vo MGU, M., 1990 | Zbl

[2] Kurant R., Metody matematicheskoi fiziki, T. 1, Gostekhizdat, M., 1951

[3] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988

[4] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976

[5] Mizokhata S., Teoriya uravnenii s chastnymi proizvodnymi, Mir, M., 1977

[6] Sukretnyi V. I., “Usrednenie kraevykh zadach dlya ellipticheskikh uravnenii v perforirovannykh oblastyakh”, UMN, 38:6 (1983), 125–126 | MR | Zbl

[7] Sukretnyi V. I., “Asimptoticheskoe razlozhenie reshenii tretei kraevoi zadachi dlya ellipticheskikh uravnenii vtorogo poryadka v perforirovannykh oblastyakh”, UMN, 39:4 (1984), 120–121 | MR

[8] Sukretnyi V. I., O zadache statsionarnoi teploprovodnosti v perforirovannykh oblastyakh, Preprint No. 84.48, Institut matematiki AN USSR, Kiev, 1984

[9] Bakhvalov N. S., Panasenko G. P., Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | Zbl

[10] Sanches-Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984