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Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 2, Pages 81–148 (Mi izv380)  

This article is cited in 68 scientific papers (total in 68 papers)

Homogenization of elasticity problems on singular structures

V. V. Zhikov

Vladimir State Pedagogical University

Abstract: We consider homogenization theory on periodic networks, junctions and more general singular objects. We show that the homogenized problem typically has a “non-classical” character. This fact is a distinctive feature of homogenization of elasticity problems in contrast to scalar problems.
We investigate the properties of Sobolev spaces for various singular structures, prove a non-classical homogenization principle for singular periodic structures of general type and describe a “scaling effect” for model problems with two small geometrical parameters.


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English version:
Izvestiya: Mathematics, 2002, 66:2, 299–365

Bibliographic databases:

UDC: 517.9
MSC: 35B27
Received: 23.11.2000
Revised: 10.09.2001

Citation: V. V. Zhikov, “Homogenization of elasticity problems on singular structures”, Izv. RAN. Ser. Mat., 66:2 (2002), 81–148; Izv. Math., 66:2 (2002), 299–365

Citation in format AMSBIB
\by V.~V.~Zhikov
\paper Homogenization of elasticity problems on singular structures
\jour Izv. RAN. Ser. Mat.
\yr 2002
\vol 66
\issue 2
\pages 81--148
\jour Izv. Math.
\yr 2002
\vol 66
\issue 2
\pages 299--365

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    This publication is cited in the following articles:
    1. Pastukhova S.E., “Homogenization of elasticity problems on periodic box structures of critical thickness”, Doklady Mathematics, 66:3 (2002), 369–373  mathscinet  zmath  isi  elib
    2. Zhikov V.V., Pastukhova S.E., “Homogenization of elasticity problems on periodic lattices of critical thickness”, Doklady Mathematics, 66:1 (2002), 94–98  mathscinet  zmath  isi  elib
    3. Pastukhova S.E., “Homogenization for nonlinear elasticity problems on thin periodic structures”, Doklady Mathematics, 65:2 (2002), 257–261  mathscinet  zmath  isi  elib
    4. V. V. Zhikov, S. E. Pastukhova, “Homogenization on periodic lattices”, Dokl. Math., 68:1 (2003), 79–83  mathnet  mathscinet  mathscinet  zmath  isi  elib
    5. Komarov A. V., Penkin O. M., Pokornyi Yu. V., “On the frequency spectrum of a multidimensional analogue of a fabric membrane”, Dokl. Math., 67:3 (2003), 323–325  mathnet  mathscinet  zmath  isi  elib
    6. Zhikov V. V., Pastukhova S. E., “Korn's inequalities for thin periodic structures”, Dokl. Math., 67:1 (2003), 55–59  mathnet  mathscinet  zmath  isi  elib
    7. V. V. Zhikov, S. E. Pastukhova, “Homogenization for elasticity problems on periodic networks of critical thickness”, Sb. Math., 194:5 (2003), 697–732  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. V. V. Shumilova, “On the Homogenization of a Problem with Two Small Parameters in Double-Porosity Media”, Math. Notes, 74:5 (2003), 753–756  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. E. M. Bogatov, O. M. Penkin, “The Fourier method in the Cauchy problem for a fourth-order equation on stratified sets”, Russian Math. (Iz. VUZ), 47:8 (2003), 65–69  mathnet  mathscinet  zmath  elib
    10. Pastukhova S. E., “On the convergence of hyperbolic semigroups in a variable space”, Dokl. Math., 70:1 (2004), 609–614  mathnet  mathscinet  zmath  isi  elib
    11. Pastukhova S. E., “Homogenization of elasticity problems on a periodic composite structure”, Dokl. Math., 69:2 (2004), 208–213  mathnet  mathscinet  zmath  isi  elib
    12. Pastukhova S. E., “Homogenization of elasticity problems on periodic rod frames of critical thickness”, Dokl. Math., 69:1 (2004), 20–25  mathnet  mathscinet  zmath  isi  elib
    13. S. E. Pastukhova, “About homogenization of elasticity problems on combined structures”, J. Math. Sci. (N. Y.), 132:3 (2006), 313–330  mathnet  crossref  mathscinet  zmath  elib  elib
    14. S. A. Nazarov, “Elliptic Boundary Value Problems in Hybrid Domains”, Funct. Anal. Appl., 38:4 (2004), 283–297  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    20. V. V. Shumilova, “Compactness principle for periodic singular and fine structures”, Math. Notes, 79:6 (2006), 878–885  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    21. V. V. Shumilova, “Compactness principle for periodic singular and fine structures”, Math. Notes, 79:2 (2006), 224–231  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    23. Shumilova V.V., “On one property of two-scale convergence”, Differ. Equ., 42:1 (2006), 155–157  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
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    25. S. E. Pastukhova, “Degenerate equations of monotone type: Lavrent'ev phenomenon and attainability problems”, Sb. Math., 198:10 (2007), 1465–1494  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    59. Bellieud M., “Homogenization of Stratified Elastic Composites With High Contrast”, SIAM J. Math. Anal., 49:4 (2017), 2615–2665  crossref  mathscinet  zmath  isi  scopus
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    67. Cherednichenko K.D., Evans J.A., “Homogenisation of Thin Periodic Frameworks With High-Contrast Inclusions”, J. Math. Anal. Appl., 473:2 (2019), 658–679  crossref  mathscinet  zmath  isi  scopus
    68. Cherednichenko K. Ershova Yu. Kiselev V A., “Time-Dispersive Behavior as a Feature of Critical-Contrast Media”, SIAM J. Appl. Math., 79:2 (2019), 690–715  crossref  mathscinet  zmath  isi
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