Homogenization of an elliptic system under condensing perforation of the domain

Nazarov, S.; Slutskiĭ, A.

doi:10.1090/S1061-0022-06-00937-X

Homogenization of an elliptic system under condensing perforation of the domain
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by S. A. Nazarov and A. S. Slutskiĭ
Translated by: A. Plotkin

St. Petersburg Math. J. 17 (2006), 989-1014

DOI: https://doi.org/10.1090/S1061-0022-06-00937-X

Published electronically: September 20, 2006

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Abstract:

Homogenization of a system of second-order differential equations is performed in the case of a nonuniformly perforated rectangle where the sizes of the holes and the distances between pairs of them decrease as the distance from one of the bases of the rectangle increases. The Neumann conditions are assumed on the boundaries of the holes. The formal asymptotics of the solution is constructed, which involves the usual Ansatz of homogenization theory and also some Ansätze typical of solutions of boundary-value problems in thin domains, in particular, exponential boundary layers. Justification of the asymptotics is done with the help of the Korn inequality, which is proved for the perforated domain $\Omega (h)$. Depending on the properties of the right-hand side, the norm of the difference between the true and the approximate solutions in the Sobolev space $H^1(\Omega (h))$ is estimated by the quantity $ch^\varkappa$ with $\varkappa \in (0,1/2]$.

References

O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoĭ fiziki, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0599579
Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
S. A. Nazarov, Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes, Uspekhi Mat. Nauk 54 (1999), no. 5(329), 77–142 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 5, 947–1014. MR 1741662, DOI 10.1070/rm1999v054n05ABEH000204
S. A. Nazarov, Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators, J. Math. Sci. (New York) 92 (1998), no. 6, 4338–4353. Some questions of mathematical physics and function theory. MR 1668418, DOI 10.1007/BF02433440
V. A. Marchenko and E. Ya. Khruslov, Kraevye zadachi v oblastyakh s melkozernistoĭ granitseĭ, Izdat. “Naukova Dumka”, Kiev, 1974 (Russian). MR 0601059
N. S. Bakhvalov and G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, “Nauka”, Moscow, 1984 (Russian). Matematicheskie zadachi mekhaniki kompozitsionnykh materialov. [Mathematical problems of the mechanics of composite materials]. MR 797571
Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
O. A. Oleĭnik, G. A. Iosif′yan, and A. S. Shamaev, Matematicheskie zadachi teorii sil′no neodnorodnykh uprugikh sred, Moskov. Gos. Univ., Moscow, 1990 (Russian). MR 1115306
V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Usrednenie differentsial′nykh operatorov, “Nauka”, Moscow, 1993 (Russian, with English and Russian summaries). MR 1318242
Doina Cioranescu and Jeannine Saint Jean Paulin, Homogenization of reticulated structures, Applied Mathematical Sciences, vol. 136, Springer-Verlag, New York, 1999. MR 1676922, DOI 10.1007/978-1-4612-2158-6
Serguei M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys. 153 (1993), no. 2, 339–357. MR 1218305
V. V. Zhikov, Connectedness and averaging. Examples of fractal conductivity, Mat. Sb. 187 (1996), no. 8, 3–40 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 8, 1109–1147. MR 1418340, DOI 10.1070/SM1996v187n08ABEH000150
S. A. Nazarov, A general scheme for averaging selfadjoint elliptic systems in multidimensional domains, including thin domains, Algebra i Analiz 7 (1995), no. 5, 1–92 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 5, 681–748. MR 1365812
—, Asymptotic theory of thin plates and rods. Dimension reduction and integral estimates. Vol. 1, “Nauchn. Kniga,” Novosibirsk, 2002. (Russian)
S. A. Nazarov and A. S. Slutskiĭ, Branching periodicity: homogenization of the Dirichlet problem for an elliptic system, Dokl. Akad. Nauk 397 (2004), no. 6, 743–747 (Russian). MR 2120175
S. A. Nazarov, Elliptic boundary value problems with periodic coefficients in a cylinder, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 101–112, 239 (Russian). MR 607578
P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37 (1982), no. 4(226), 3–52, 240 (Russian). MR 667973
S. A. Nazarov, Nonselfadjoint elliptic problems with the polynomial property in domains possessing cylindrical outlets to infinity, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 249 (1997), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 29, 212–230, 316–317 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 101 (2000), no. 5, 3512–3522. MR 1698519, DOI 10.1007/BF02680148
Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387, DOI 10.1515/9783110848915.525