Geodesic
Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain
Algebra i analiz, Tome 21 (2009) no. 4, pp. 126-173 
G. Cardone; A. Corbo Esposito; S. A. Nazarov. Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain. Algebra i analiz, Tome 21 (2009) no. 4, pp. 126-173. http://geodesic.mathdoc.fr/item/AA_2009_21_4_a2/
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     author = {G. Cardone and A. Corbo Esposito and S. A. Nazarov},
     title = {Homogenization of the mixed boundary value problem for a~formally self-adjoint system in a~periodically perforated domain},
     journal = {Algebra i analiz},
     pages = {126--173},
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     volume = {21},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2009_21_4_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A generalized Gårding-Korn inequality is established in a domain $\Omega(h)\subset{\mathbb{R}}^n$ with a small, of size $O(h)$, periodic perforation, without any restrictions on the shape of the periodicity cell, except for the usual assumptions that the boundary is Lipschitzian, which ensures the Korn inequality in a general domain. Homogenization is performed for a formally selfadjoint elliptic system of second order differential equations with the Dirichlet or Neumann conditions on the outer or inner parts of the boundary, respectively; the data of the problem are assumed to satisfy assumptions of two types: additional smoothness is required from the dependence on either the “slow” variables $x$, or the “fast” variables $y=h^{-1}x$. It is checked that the exponent $\delta\in(0,1/2]$ in the accuracy $O(h^\delta)$ $O(h^\delta)$ of homogenization depends on the smoothness properties of the problem data.

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

[2] Nečas J., Les méthodes directes en théorie des équations elliptiques, Masson, Paris; Academia, Prague, 1967 | MR | Zbl

[3] Nazarov S. A., “Samosopryazhënnye ellipticheskie kraevye zadachi. Polinomialnoe svoistvo i formalno polozhitelnye operatory”, Probl. mat. anal., 16, SPbGU, SPb., 1997, 167–192

[4] Nazarov S. A., “Polinomialnoe svoistvo samosopryazhënnykh ellipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov”, Uspekhi mat. nauk, 54:5 (1999), 77–142 | MR | Zbl

[5] Lekhnitskii S. G., Teoriya uprugosti anizotropnogo tela, Nauka, M., 1977 | MR | Zbl

[6] Nazarov S. A., Asimptoticheskaya teoriya tonkikh plastin i sterzhnei. Ponizhenie razmernosti i integralnye otsenki, T. 1, Nauch. kn., Novosibirsk, 2002

[7] Parton V. Z., Kudryavtsev B. A., Elektrouprugost pezoelektricheskikh i elektroprovodnykh tel, Nauka, M., 1988

[8] Nazarov S. A., “Ravnomernye otsenki ostatkov v asimptoticheskikh razlozheniyakh resheniya zadachi o kolebaniyakh pezoelektricheskoi plastiny”, Probl. mat. anal., 25, Tamara Rozhkovskaya, Novosibirsk, 2003, 99–188

[9] Akimova E. A., Nazarov S. A., Chechkin G. A., “Vesovoe neravenstvo Korna: protsedura “tetris”, obsluzhivayuschaya proizvolnuyu periodicheskuyu plastinu”, Dokl. RAN, 380:4 (2001), 439–442 | MR | Zbl

[10] Akimova E. A., Nazarov S. A., Chechkin G. A., “Asimptotika resheniya zadachi deformatsii proizvolnoi lokalno periodicheskoi tonkoi plastiny”, Tr. Mosk. mat. o-va, 65, 2004, 3–34 | MR | Zbl

[11] Nazarov S. A., “Neravenstva Korna dlya uprugikh sochlenenii massivnykh tel, tonkikh plastin i sterzhnei”, Uspekhi mat. nauk, 63:1 (2008), 37–110 | MR | Zbl

[12] Nazarov S. A., “Obosnovanie asimptoticheskoi teorii tonkikh sterzhnei. Integralnye i potochechnye otsenki”, Probl. mat. anal., 17, SPbGU, SPb., 1997, 101–152

[13] Bensoussan A., Lions J.-L., Papanicolaou G., Asymptotic analysis for periodic structures, Stud. in Math. Appl., 5, North-Holland, Amsterdam–New York, 1978 | MR | Zbl

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

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

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

[17] Nazarov S. A., “Otsenki vesovykh $L_p$-norm i maksimuma modulya asimptoticheskikh ostatkov pri osrednenii ellipticheskikh sistem s periodicheskimi koeffitsientami”, Algebra i analiz, 18:2 (2006), 117–166 | MR | Zbl

[18] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[19] Birman M. Sh., Suslina T. A., “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva i usredneniya”, Algebra i analiz, 15:5 (2003), 1–108 | MR | Zbl

[20] Birman M. Sh., Suslina T. A., “Usrednenie periodicheskikh ellipticheskikh differentsialnykh operatorov s uchetom korrektora”, Algebra i analiz, 17:6 (2005), 1–104 | MR

[21] Birman M. Sh., Suslina T. A., “Usrednenie periodicheskikh ellipticheskikh differentsialnykh operatorov s uchetom korrektora. Priblizhenie reshenii v klasse Soboleva $H^1(\mathbb R^d)$”, Algebra i analiz, 18:6 (2006), 1–130 | MR | Zbl

[22] Zhikov V. V., “O spektralnom metode v teorii usredneniya”, Tr. Mat. in-ta RAN, 250, 2005, 95–104 | MR | Zbl

[23] Zhikov V. V., Pastukhova S. E., “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524 | MR | Zbl

[24] Pastukhova S. E., “O nekotorykh otsenkakh iz usredneniya zadach teorii uprugosti”, Dokl. RAN, 406:5 (2006), 604–608 | MR | Zbl

[25] Agmon S., Douglis A., Nirenberg L., “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II”, Comm. Pure Appl. Math., 17 (1964), 35–92 | DOI | MR | Zbl

[26] Solonnikov V. A., “Ob obschikh kraevykh zadachakh dlya sistem, ellipticheskikh v smysle A. Duglisa–L. Nirenberga. I”, Izv. AN SSSR. Ser. mat., 28:3 (1964), 665–706 ; “II”, Тр. Мат. ин-та АН СССР, 92, 1966, 233–297 | MR | Zbl | MR | Zbl