Averaging in a perforated domain with an oscillating third boundary condition

A. G. Belyaev   ; A. L. Piatnitski   ; G. A. Chechkin

Sbornik. Mathematics, Tome 192 (2001) no. 7, pp. 933-949

Résumé

We study an example averaging problem for a second-order elliptic equation in a periodically perforated domain with a third boundary condition (Fourier condition) on the boundary of the holes. Under the assumption that the coefficients of the boundary operator are bounded and the corresponding averages are small we construct the leading terms of the asymptotic expansion of the solution and estimate the error.

Bibliographie

[1] Homogenization and porous media, ed. Hornung U., Springer-Verlag, Berlin, 1997 | MR

[2] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993 | MR | Zbl

[3] Marchenko V. A., Khruslov E. Ya., Kraevye zadachi v oblastyakh s melkozernistoi granitsei, Naukova dumka, Kiev, 1974 | MR

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

[5] Sanchez-Palencia E., Homogenization techniques for composite media, Springer-Verlag, Berlin, 1987 | MR | Zbl

[6] Saint Jean Paulin J., Cioranescu D., Homogenization of reticulated structures, Appl. Math. Sci., 136, Springer-Verlag, Berlin, 1999 | MR

[7] Cioranescu D., Murat F., “Un terme étrange venu d'ailleurs”, Nonlinear partial differential equations and their applications, Res. Notes in Math., 60, Pitman, London, 1982, 98–138 | MR

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

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

[10] Sukretnyi V. I., O zadache statsionarnoi teploprovodnosti v perforirovannykh oblastyakh, Preprint 84.48, In-t matematiki AN USSR, Kiev, 1984 | MR

[11] Pastukhova S. E., “Usrednenie smeshannoi zadachi s kosoi proizvodnoi dlya ellipticheskogo operatora v perforirovannoi oblasti”, Differents. uravneniya, 30:8 (1994), 1445–1456 | MR | Zbl

[12] Pastukhova S. E., “O pogreshnosti usredneniya dlya zadachi Steklova v perforirovannoi oblasti”, Differents. uravneniya, 31:6 (1995), 1042–1054 | MR

[13] Cioranescu D., Donato P., “Homogénéisation du problème de Neumann non homogène dans des ouverts perforés”, Asymptotic Anal., 1 (1988), 115–138 | MR | Zbl

[14] Belyaev A. G., Pyatnitskii A. L., Chechkin G. A., “Asimptoticheskoe povedenie resheniya kraevoi zadachi v perforirovannoi oblasti s ostsilliruyuschei granitsei”, Sib. matem. zhurn., 39:4 (1998), 730–754 | MR | Zbl

[15] Chechkin G. A., Piatnitski A. L., “Homogenization of boundary-value problem in a locally periodic perforated domain”, Appl. Anal., 71:1–4 (1999), 215–235 | DOI | MR | Zbl

[16] Pastukhova S. E., “Metod kompensirovannoi kompaktnosti Tartara v usrednenii spektra smeshannoi zadachi dlya ellipticheskogo uravneniya v perforirovannoi oblasti s tretim kraevym usloviem”, Matem. sb., 186:5 (1995), 127–144 | MR | Zbl

[17] Pastukhova S. E., “O kharaktere raspredeleniya polya temperatur v perforirovannom tele s zadannym ego znacheniem na vneshnei granitse v usloviyakh teploobmena na granitse polostei po zakonu Nyutona”, Matem. sb., 187:6 (1996), 85–96 | MR | Zbl

[18] Pastukhova S. E., “Obosnovanie asimptotiki resheniya smeshannoi zadachi dlya statsionarnoi teploprovodnosti v perforirovannoi oblasti”, Tr. MMO, 58, URSS, M., 1997, 88–101 | MR | Zbl

[19] Volkov D. B., “Ob osrednenii nekotorykh kraevykh zadach v oblastyakh s periodicheskoi strukturoi”, ZhVM i MF, 22:1 (1982), 112–122 | MR | Zbl

[20] Cioranescu D., Saint Jean Paulin J., “Truss structures: Fourier conditions and eigenvalue problems”, Boundary control and boundary variation, Lecture Notes Control Inf. Sci., 178, ed. J. P. Zolezio, Springer-Verlag, Berlin, 1992, 125–141 | MR

[21] Cioranescu D., Donato P., “On a Robin problem in perforated domains”, Homogenization and applications to material sciences, GAKUTO International Series. Mathematical Sciences and Applications, 9, eds. D. Cioranescu, A. Damlamian, P. Donato, Gakkōtosho, Tokyo, 1997, 123–135 | MR

[22] Murat F., Tartar L., Calcul des variations et homogénéisation, R 84012, Université Pierre et Marie Curie, Centre National de la Recherche Scientifique, Laboratoire d'analyse numérique, Paris, 1984 | MR

[23] Topics in mathematical modelling of composite materials, eds. Cherkaev A., Kohn R., Birkhäuser, Boston, 1997 | MR

[24] Nguetseng G., “A general convergence result for a functional related to the theory of homogenization”, SIAM J. Math. Anal., 20:3 (1989), 608–623 | DOI | MR | Zbl

[25] Allaire G., “Homogenization and two-scale convergence”, SIAM J. Math. Anal., 23 (1992), 1482–1518 | DOI | MR | Zbl

[26] Neuss-Radu M., “Some extensions of two-scale convergence”, C. R. Acad. Sci. Paris Sér. I Math., 322:9 (1996), 899–904 | MR | Zbl

[27] Bakhvalov N. S., “Usrednennye kharakteristiki tel s periodicheskoi strukturoi”, Dokl. AN SSSR, 218:5 (1974), 1046–1048 | MR

[28] Bakhvalov N. S., “Usrednenie differentsialnykh uravnenii s chastnymi proizvodnymi s bystro ostsilliruyuschimi koeffitsientami”, Dokl. AN SSSR, 221:3 (1975), 516–519 | MR | Zbl

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

[30] Chechkin G. A., Friedman A., Piatnitski A. L., “The boundary-value problem in domains with very rapidly oscillating boundary”, J. Math. Anal. Appl., 231:1 (1999), 213–234 | DOI | MR | Zbl