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Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 5, Pages 961–998 (Mi izv1433)  

This article is cited in 25 scientific papers (total in 25 papers)

Questions of convergence, duality, and averaging for functionals of the calculus of variations

V. V. Zhikov


Abstract: The concept of $\Gamma$-convergence is studied for functionals of the calculus of variations (this concept was introduced and studied by Italian mathematicians in the school of De Giorgi), the concept of $\Gamma$-convergence is introduced and studied for dual functionals, and a duality theory is constructed connecting the $\Gamma$-limits of the original and the dual functionals. The problem of integral representation and the averaging problem are considered on the basis of this. Some unsolved problems are formulated.
Bibliography: 22 titles.

Full text: PDF file (3696 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1984, 23:2, 243–276

Bibliographic databases:

UDC: 519.3
MSC: Primary 49A50, 49A55, 54A20; Secondary 34C29, 34C35, 35J20, 46E30, 46E35, 52A99, 54C65, 5
Received: 23.08.1982

Citation: V. V. Zhikov, “Questions of convergence, duality, and averaging for functionals of the calculus of variations”, Izv. Akad. Nauk SSSR Ser. Mat., 47:5 (1983), 961–998; Math. USSR-Izv., 23:2 (1984), 243–276

Citation in format AMSBIB
\Bibitem{Zhi83}
\by V.~V.~Zhikov
\paper Questions of convergence, duality, and averaging for functionals of the calculus of variations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1983
\vol 47
\issue 5
\pages 961--998
\mathnet{http://mi.mathnet.ru/izv1433}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=718413}
\zmath{https://zbmath.org/?q=an:0551.49012|0534.49014}
\transl
\jour Math. USSR-Izv.
\yr 1984
\vol 23
\issue 2
\pages 243--276
\crossref{https://doi.org/10.1070/IM1984v023n02ABEH001466}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Math. USSR-Izv., 29:1 (1987), 33–66  mathnet  crossref  mathscinet  zmath
    2. Berdichevskii V., “The Problem of Averaging Random Structures in Terms of Distribution-Functions”, Pmm-J. Appl. Math. Mech., 51:6 (1987), 704–711  crossref  mathscinet  isi
    3. S. M. Kozlov, “Geometric aspects of averaging”, Russian Math. Surveys, 44:2 (1989), 91–144  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. V. V. Zhikov, “On passage to the limit in nonlinear variational problems”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 427–459  mathnet  crossref  mathscinet  zmath  isi
    5. O. O. Barabanov, V. V. Zhikov, “The limit load and homogenization”, Russian Acad. Sci. Izv. Math., 43:2 (1994), 205–231  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. V. V. Zhikov, “To the Problem of Passage to the Limit in Divergent Nonuniformly Elliptic Equations”, Funct. Anal. Appl., 35:1 (2001), 19–33  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Yu. A. Alkhutov, O. V. Krasheninnikova, “Continuity at boundary points of solutions of quasilinear elliptic equations with a non-standard growth condition”, Izv. Math., 68:6 (2004), 1063–1117  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. Yu. A. Alkhutov, “Hölder continuity of $p(x)$-harmonic functions”, Sb. Math., 196:2 (2005), 147–171  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. V. V. Zhikov, S. E. Pastukhova, “Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent”, Sb. Math., 199:12 (2008), 1751–1782  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Yu. A. Alkhutov, O. V. Krasheninnikova, “On the Continuity of Solutions to Elliptic Equations with Variable Order of Nonlinearity”, Proc. Steklov Inst. Math., 261 (2008), 1–10  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    11. S. E. Pastukhova, A. S. Khripunova, “Gamma–closure of some classes of nonstandard convex integrands”, J Math Sci, 2011  crossref
    12. Yu. Alkhutov, “Elliptic problems with nonstandard conditions of growth: Zhikov's approach”, Complex Variables and Elliptic Equations, 56:7-9 (2011), 559  crossref
    13. V. V. Zhikov, S. E. Pastukhova, “The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions”, Sb. Math., 205:4 (2014), 488–521  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. V. V. Zhikov, S. E. Pastukhova, “On integral representation of $\Gamma$-limit functionals”, J. Math. Sci., 217:6 (2016), 736–750  mathnet  crossref  mathscinet
    15. Kovalevsky A.A., “On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains”, Nonlinear Anal.-Theory Methods Appl., 147 (2016), 63–79  crossref  mathscinet  zmath  isi  scopus
    16. A. A. Kovalevskii, “Variatsionnye zadachi s odnostoronnimi potochechno funktsionalnymi ogranicheniyami v peremennykh oblastyakh”, Tr. IMM UrO RAN, 23, no. 2, 2017, 133–150  mathnet  crossref  elib
    17. V. G. Osmolovskiǐ, “Mathematical problems in the theory of phase transitions in continuum mechanics”, St. Petersburg Math. J., 29:5 (2018), 793–839  mathnet  crossref  mathscinet  isi  elib
    18. Alexander A. Kovalevsky, “Convergence of solutions of bilateral problems in variable domains and related questions”, Ural Math. J., 3:2 (2017), 51–66  mathnet  crossref
    19. Alkhutov Yu.A. Surnachev M.D., “On the Regularity of a Boundary Point For the P(X)-Laplacian”, Dokl. Math., 97:1 (2018), 65–68  crossref  isi
    20. A. A. Kovalevsky, “On the Convergence of Solutions of Variational Problems with Implicit Pointwise Constraints in Variable Domains”, Funct. Anal. Appl., 52:2 (2018), 147–150  mathnet  crossref  crossref  isi  elib
    21. V. V. Zhikov, S. E. Pastukhova, “Usrednenie i dvukhmasshtabnaya skhodimost v sobolevskom prostranstve s ostsilliruyuschim pokazatelem”, Algebra i analiz, 30:2 (2018), 114–144  mathnet  elib
    22. M. D. Surnachev, “O neravenstve Kharnaka dlya $p(x)$-laplasiana”, Preprinty IPM im. M. V. Keldysha, 2018, 069, 32 pp.  mathnet  crossref
    23. A. A. Kovalevskii, “O skhodimosti reshenii variatsionnykh zadach s neyavnymi ogranicheniyami, zadannymi bystro ostsilliruyuschimi funktsiyami”, Tr. IMM UrO RAN, 24, no. 2, 2018, 107–122  mathnet  crossref  elib
    24. Alkhutov Yu.A. Surnachev M.D., “A Harnack Inequality For a Transmission Problem With P(X)-Laplacian”, Appl. Anal., 98:1-2, SI (2019), 332–344  crossref  mathscinet  isi  scopus
    25. Yu. A. Alkhutov, M. D. Surnachev, “Povedenie v granichnoi tochke reshenii zadachi Dirikhle dlya $p(x)$-laplasiana”, Algebra i analiz, 31:2 (2019), 88–117  mathnet
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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