<i>Γ</i>-convergence of nonlinear functionals in thin reticulated structures

Leonid Pankratov

doi:10.1016/S1631-073X(02)02468-8

Γ-convergence of nonlinear functionals in thin reticulated structures
[Γ-convergence des fonctionelles non linéaires dans des structures réticulées de faible épaisseur]

Leonid Pankratov ¹

¹ Département de mathématiques, Institut des Basses Températures (FTINT), 47, av. Lénine, 61164 Kharkov, Ukraine

Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 315-320.

Résumé
Abstract

On étudie la Γ-convergence de fonctionelles non linéaires considérées dans des structures non périodiques de type de grille dans l'espace $R^{2}$ . La fonctionelle Γ-limite est obtenue sous forme explicite.

We study the Γ-convergence of nonlinear functionals considered in nonperiodic 2D lattice-like structures. The Γ-limit functional is obtained in the explicit form.

Reçu le : 2001-05-14
Accepté le : 2002-06-03
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02468-8

Affiliations des auteurs :

Leonid Pankratov ¹

¹ Département de mathématiques, Institut des Basses Températures (FTINT), 47, av. Lénine, 61164 Kharkov, Ukraine

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Leonid Pankratov. Γ-convergence of nonlinear functionals in thin reticulated structures. Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 315-320. doi : 10.1016/S1631-073X(02)02468-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02468-8/

Bibliographie
Cité par

[1] N.S. Bakhvalov; G.P. Panasenko Homogenization: Averaging Processes in Periodic Media, Kluwer Academic, Dordrecht, 1989

[2] A. Bensoussan; J.-L. Lions; G. Papanicolau Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., 5, North-Holland, Amsterdam, 1978

[3] A. Braides; A. Defranceschi Homogenization of Multiple Integrals, Oxford Lecture Ser. Math. Appl., 12, Clarendon Press, Oxford, 1998

[4] D. Cioranescu; J. Saint Jean Paulin Homogenization of Reticulated Structures, Appl. Math. Sci., 136, Springer-Verlag, New York, 1999

[5] D. Cioranescu, M.V. Goncharenko, F. Murat, L.S. Pankratov, Homogenization of nonlinear variational problems in domains of degenerating measure, prepared for publication

[6] G. Dal Maso An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993

[7] E. Ya Khruslov; L.S. Pankratov Homogenization of the Dirichlet variational problems in Orlicz–Sobolev spaces, Oper. Theory Appl., Fields Inst. Commun., 25, American Mathematical Society, Providence, RI, 2000, pp. 345-366

[8] A.A. Kovalevskij Conditions of the Γ-convergence and homogenization of integral functionals with different domains of the definition, Dokl. Akad. Nauk Ukrainy, Volume 4 (1991), pp. 5-8 (in Russian)

[9] O.A. Ladyzhenskaya; N.N. Ural'tseva Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1973

[10] L.S. Pankratov, On convergence of the solutions of variational problems in weakly connected domains, Preprint 53-88, Institute for Low Temperature Physics and Engineering, Kharkov, 1988 (in Russian)

[11] É. Sanchez-Palencia Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys., 127, Springer-Verlag, New York, 1980

[12] E.V. Svischeva Asymptotic behavior of the solutions of the second boundary value problem in domains of decreasing volume, Operator Theory and Subharmonic Functions, Naukova Dumka, Kiev, 1991, pp. 126-134 (in Russian)

[13] V.V. Zhikov; S.M. Kozlov; O.A. Oleinik Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, New York, 1994

Alexander A. Kovalevsky Approximation in $W^{1, p}$ -norms of solutions of minimum problems with bilateral constraints in variable domains, Bollettino dell'Unione Matematica Italiana (2025) | DOI:10.1007/s40574-025-00467-6
Alexander A. Kovalevsky Convergence of solutions of variational problems with measurable bilateral constraints in variable domains, Annali di Matematica Pura ed Applicata. Serie Quarta, Volume 201 (2022) no. 2, pp. 835-859 | DOI:10.1007/s10231-021-01140-3 | Zbl:1485.49017
Alexander A. Kovalevsky On the convergence of solutions of variational problems with pointwise functional constraints in variable domains, Journal of Mathematical Sciences (New York), Volume 254 (2021) no. 3, pp. 375-396 | DOI:10.1007/s10958-021-05310-9 | Zbl:1465.49012
Alexander Kovalevsky On the convergence of solutions of variational problems with pointwise functional constraints in variable domains, Ukrainian Mathematical Bulletin, Volume 17 (2020) no. 4, p. 509 | DOI:10.37069/1810-3200-2020-17-4-3
Alexander A. Kovalevsky Variational problems with variable regular bilateral constraints in variable domains, Revista Matemática Complutense, Volume 32 (2019) no. 2, pp. 327-351 | DOI:10.1007/s13163-018-0281-6 | Zbl:1419.49015
A. A. Kovalevsky Variational problems with unilateral pointwise functional constraints in variable domains, Proceedings of the Steklov Institute of Mathematics, Volume 301 (2018), p. s115-s131 | DOI:10.1134/s0081543818050097 | Zbl:1401.49029
D. Trucu; M. A. J. Chaplain; A. Marciniak-Czochra Three-scale convergence for processes in heterogeneous media, Applicable Analysis, Volume 91 (2012) no. 7, pp. 1351-1373 | DOI:10.1080/00036811.2011.569498 | Zbl:1252.35038
O. A. Rudakova On Γ-convergence of integral functionals defined on various weighted Sobolev spaces, Ukrainian Mathematical Journal, Volume 61 (2009) no. 1, p. 121 | DOI:10.1007/s11253-009-0193-1
B. Amaziane; M. Goncharenko; L. Pankratov $Γ_{D}$ -convergence for a class of quasilinear elliptic equations in thin structures, Mathematical Methods in the Applied Sciences, Volume 28 (2005) no. 15, pp. 1847-1865 | DOI:10.1002/mma.644 | Zbl:1185.35012

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