Asymptotic analysis of boundary-value problems in thick three-dimensional multi-level junctions

T. A. Mel'nik   ; G. A. Chechkin

Sbornik. Mathematics, Tome 200 (2009) no. 3, pp. 357-383

Résumé

We consider homogenization problems in a singularly perturbed three-dimensional domain of multi-level-junction type which consists of the junction body and a large number of alternating thin curvilinear cylinders that belong to two classes. Under the assumption that one class consists of cylinders of finite height, and the second class of cylinders of infinitesimal height, and that different inhomogeneous boundary conditions of the third kind with perturbed coefficients are given on the boundaries of the thin curvilinear cylinders, we prove the homogenization theorems and the convergence of the energy integrals. Bibliography: 42 titles.

Détail

Keywords: homogenization, thick junctions, asymptotics.

Bibliographie

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

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

[3] N. S. Bakhvalov, G. Panasenko, Homogenisation: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, Math. Appl. (Soviet Ser.), 36, Kluwer Acad. Publ., Dordrecht, 1989 | MR | MR | Zbl | Zbl

[4] V. V. Jikov, S. M. Kozlov, O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994 | MR | MR | Zbl | Zbl

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

[6] S. A. Nazarov, Asimptoticheskaya teoriya tonkikh plastin i sterzhnei, Nauchnaya kniga, Novosibirsk, 2002

[7] G. A. Chechkin, A. L. Piatnitski, A. S. Shamaev, Homogenization. Methods and applications, Transl. Math. Monogr., 234, Amer. Math. Soc., Providence, RI, 2007 | MR | Zbl

[8] D. Blanchard, A. Gaudiello, T. A. Mel'nyk, “Boundary homogenization and reduction of dimension in a Kirchhoff–Love plate”, SIAM J. Math. Anal., 39:6 (2008), 1764–1787 | DOI | MR | Zbl

[9] U. De Maio, T. Durante, T. A. Mel'nyk, “Asymptotic approximation for the solution to the Robin problem in a thick multi-level junction”, Math. Models Methods Appl. Sci., 15:12 (2005), 1897–1921 | DOI | MR | Zbl

[10] F. Fleury, E. Sánchez-Palencia, “Asymptotics and spectral properties of the acoustic vibrations of a body perforated by narrow channels”, Bull. Sci. Math. (2), 110:2 (1986), 149–176 | MR | Zbl

[11] T. A. Mel'nik, S. A. Nazarov, “The asymptotic structure of the spectrum in the problem of harmonic oscillations of a hub with heavy spokes”, Russian Acad. Sci. Dokl. Math., 48:3 (1994), 428–432 | MR | Zbl

[12] T. A. Mel'nik, S. A. Nazarov, “The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type”, J. Math. Sci. (New York), 85:6 (1997), 2326–2346 | DOI | MR | Zbl

[13] T. A. Mel'nyk, “Homogenization of the Poisson equation in a thick periodic junction”, Z. Anal. Anwendungen, 18:4 (1999), 953–975 | MR | Zbl

[14] T. A. Mel'nik, S. A. Nazarov, “Asymptotic analysis of the Neumann problem on the junction of a body and thin heavy rods”, St. Petersburg Math. J., 12:2 (2001), 317–351 | MR | Zbl

[15] T. A. Mel'nik, “Homogenization of a singularly perturbed parabolic problem in a thick periodic junction of the type 3:2:1”, Ukr. matem. zhurn., 52:11 (2000), 1524–1533 | MR | Zbl

[16] T. A. Mel'nyk, “Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction”, Ukr. matem. zhurn., 58:2 (2006), 195–216 | MR | Zbl

[17] T. A. Mel'nyk, P. S. Vashchuk, “Homogenization of the Neumann–Fourier problem in a thick two-level junction of type 3:2:1.”, Zh. Mat. Fiz. Anal. Geom., 2:3 (2006), 318–337 | MR | Zbl

[18] T. A. Mel'nyk, P. S. Vashchuk, “Homogenization of a boundary value problem with mixed type of boundary conditions in a thick junction”, Differ. Equ., 43:5 (2007), 696–703 | DOI | MR | Zbl

[19] T. A. Mel'nyk, “Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1”, Math. Methods Appl. Sci., 31:9 (2008), 1005–1027 | DOI | MR | Zbl

[20] S. A. Nazarov, “Soedineniya singulyarno vyrozhdayuschikhsya oblastei razlichnykh predelnykh razmernostei. I”, Tr. sem. im. I. G. Petrovskogo, 18, Izd-vo Mosk. un-ta, M., 1995, 3–78 ; “Соединения сингулярно вырождающихся областей различных предельных размерностей. II”, 20, 1997, 155–195 ; S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions. I”, J. Math. Sci., 80:5 (1996), 1989–2034 ; “Junctions of singularly degenerating domains with different limit dimensions. II.”, J. Math. Sci. (New York), 97:3 (1999), 4085–4108 | MR | Zbl | MR | Zbl | DOI | DOI

[21] E. Ya. Khruslov, “O rezonansnykh yavleniyakh v odnoi zadache difraktsii”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, vyp. 6, Izd-vo Khark. un-ta, Kharkov, 1968, 111–129

[22] Y. Amirat, G. A. Chechkin, R. R. Gadyl'shin, “Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary”, ZhVM i MF, 46:1 (2006), 102–115 | MR | Zbl

[23] Y. Amirat, G. A. Chechkin, R. R. Gadyl'shin, “Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues”, Appl. Anal., 86:7 (2007), 873–897 | DOI | MR | Zbl

[24] A. G. Belyaev, O singulyarnykh vozmuscheniyakh granichnykh zadach, Dis. ... kand. fiz.-matem. nauk, MGU, M., 1990

[25] A. G. Belyaev, A. L. Pyatnitskiǐ, G. A. Chechkin, “Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary”, Siberian Math. J., 39:4 (1998), 621–644 | DOI | MR | Zbl

[26] G. Bouchitté, A. Lidouh, P. Suquet, “Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non lineaire et un corps rigide”, C. R. Acad. Sci. Paris Sér. I Math., 313:13 (1991), 967–972 | MR | Zbl

[27] G. Chechkin, D. Cioranescu, “Vibration of a thin plate with a “rough” surface”, Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. XIV (Paris, 1997–1998), Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002, 147–169 | DOI | MR | Zbl

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

[29] E. N. Dancer, D. Daners, “Domain perturbation for elliptic equations subject to robin boundary conditions”, J. Differential Equations, 138:1 (1997), 86–132 | DOI | MR | Zbl

[30] A. Gaudiello, “Asymptotic behaviour of non-homogeneous Neumann problems in domains with oscillating boundary”, Ricerche Mat., 43:2 (1994), 239–292 | MR | Zbl

[31] W. Jäger, A. Mikelić, “On the roughness-induced effective boundary conditions for an incompressible viscous flow”, J. Differential Equations, 170:1 (2001), 96–122 | DOI | MR | Zbl

[32] W. Jäger, A. Mikelić, “Couette flows over a rough boundary and drag reduction”, Comm. Math. Phys., 232:3 (2003), 429–455 | DOI | MR | Zbl

[33] W. Kohler, G. Papanicolaou, S. Varadhan, “Boundary and interface problems in regions with very rough boundaries”, Multiple scattering and waves in random media, Proceedings of the U.S. Army Workshop (Blacksburg, VA, 1980), North-Holland, Amsterdam–New York–Oxford, 1981, 165–197 | MR | Zbl

[34] J. Nevard, J. B. Keller, “Homogenization of rough boundaries and interfaces”, SIAM J. Appl. Math., 57:6 (1997), 1660–1686 | DOI | MR | Zbl

[35] T. A. Melnik, “Userednenie ellipticheskikh uravnenii, opisyvayuschikh protsessy v silno neodnorodnykh tonkikh perforirovannykh oblastyakh s bystro izmenyayuscheisya tolschinoi”, Dokl. AN Ukrainy, 10 (1991), 15–18 | MR

[36] G. A. Chechkin, T. P. Chechkina, “On homogenization of problems in domains of the “infusorium” type”, J. Math. Sci. (New York), 120:3 (2004), 1470–1482 | DOI | MR | Zbl

[37] G. A. Chechkin, T. P. Chechkina, “Homogenization theorem for problems in domains of the “infusorian” type with uncoordinated structure”, J. Math. Sci. (N. Y.), 123:5 (2004), 4363–4380 | DOI | MR | Zbl

[38] T. A. Melnik, G. A. Chechkin, “Asimptoticheskii analiz kraevykh zadach v gustykh kaskadnykh soedineniyakh”, Dokl. NAN Ukrainy, 9 (2008), 16–22

[39] O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci., 49, Springer–Verlag, New York, 1985 | MR | MR | Zbl | Zbl

[40] Z. Denkowski, S. Mortola, “Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations to partial differential equations”, J. Optim. Theory Appl., 78:2 (1993), 365–391 | DOI | MR | Zbl

[41] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988 ; S. L. Sobolev, Some applications of functional analysis in mathematical physics, Transl. Math. Monogr., 90, Amer. Math. Soc., Providence, RI, 1991 | MR | Zbl | MR | Zbl

[42] S. L. Sobolev, Izbrannye voprosy teorii funktsionalnykh prostranstv i obobschennykh funktsii, Nauka, M., 1989 | MR | Zbl