Two-scale div-curl lemma

Visintin, Augusto

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 2, pp. 291-321.

The div-curl lemma, one of the basic results of the theory of compensated compactness of Murat and Tartar, does not take over to the case in which the two factors two-scale converge in the sense of Nguetseng. A suitable modification of the differential operators however allows for this extension. The argument follows the lines of a well-known paper of F. Murat of 1978, and uses a two-scale extension of the Fourier transform. This result is also extended to time-dependent functions, and is applied to a two-scale formulation of the Maxwell system of electromagnetism, that accounts for the energy embedded in both coarse- and fine-scale oscillations.

[1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), 1482-1518. | MR | Zbl

[2] G. Allaire, “Shape Optimization by the Homogenization Method”, Springer, New York, 2002. | MR | Zbl

[3] T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21 (1990), 823-836. | MR | Zbl

[4] J.-P. Aubin, Un théorème de compacité, C.R. Acad. Sci. Paris, Série I 256 (1963), 5042-5044. | MR | Zbl

[5] N. Bakhvalov and G. Panasenko, “Homogenisation: Averaging Processes in Periodic Media”, Kluwer, Dordrecht, 1989. | MR | Zbl

[6] G. Bensoussan, J..L. Lions and G. Papanicolaou, “Asymptotic Analysis for Periodic Structures”, North-Holland, Amsterdam, 1978. | MR | Zbl

[7] A. Bourgeat, S. Luckhaus and A. Mikelić, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow, SIAM J. Math. Anal. 27 (1996), 1520-1543. | MR | Zbl

[8] H. Brezis, “Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert”, North-Holland, Amsterdam, 1973. | MR | Zbl

[9] M. Briane and J. Casado-Díaz, Lack of compactness in two-scale convergence, SIAM J. Math. Anal. 37 (2005), 343-346. | MR | Zbl

[10] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris, Sér. I 335 (2002), 99-104. | MR | Zbl

[11] D. Cioranescu and P. Donato, “An Introduction to Homogenization”, Oxford Univ. Press, New York, 1999. | MR | Zbl

[12] I. Ekeland and R. Temam, “Analyse Convexe et Problèmes Variationnelles”, Dunod Gauthier-Villars, Paris, 1974. | MR | Zbl

[13] J.-B. Hiriart-Urruty and C. Lemarechal, “Convex Analysis and Optimization Algorithms”, Springer, Berlin, 1993.

[14] J. D. Jackson, “Classical Electrodynamics”, Wiley, Chichester, 1962. | MR | Zbl

[15] V. V. Jikov, S. M. Kozlov and O.A. Oleinik, “Homogenization of Differential Operators and Integral Functionals”, Springer, Berlin, 1994. | MR | Zbl

[16] M. Lenczner, Homogénéisation d'un circuit électrique, C.R. Acad. Sci. Paris, Ser. II 324 (1997), 537-542. | Zbl

[17] J. L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires”, Dunod, Paris, 1969. | Zbl

[18] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 489-507. | EuDML | Numdam | MR | Zbl

[19] F. Murat, Compacité par compensation. II, In: “Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis” held in Rome in 1978, E. De Giorgi and E. Magenes (eds.), Pitagora, Bologna, 1979, 245-256. | MR | Zbl

[20] F. Murat, A survey on compensated compactness, In: “Contributions to Modern Calculus of Variations”, Bologna, 1985, Pitman Res. Notes Math. Ser., Vol. 148, Longman, Harlow, 1987, 145-183. | MR

[21] F. Murat and L. Tartar, H-convergence, In: “Topics in the Mathematical Modelling of Composite Materials”, A. Cherkaev and R. Kohn (eds.), Birkhäuser, Boston 1997, 21-44. | MR | Zbl

[22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608-623. | MR | Zbl

[23] G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics, SIAM J. Math. Anal. 21 (1990), 1394-1414. | MR | Zbl

[24] E. Sanchez-Palencia, “Non-Homogeneous Media and Vibration Theory”, Springer, New York, 1980. | MR | Zbl

[25] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. 146 (1987), 65-96. | MR | Zbl

[26] L. Tartar, “Course Peccot”, Collège de France, Paris 1977, unpublished, partially written in [21].

[27] L. Tartar, Une nouvelle méthode de résolution d'équations aux dérivées partielles non linéaires, In: “Journées d'Analyse Non linéaire”, Springer, Berlin, 1978, 228-241. | MR | Zbl

[28] L. Tartar, Compensated compactness and applications to partial differential equations, In: “Nonlinear Analysis and Mechanics: Heriott-Watt Symposium”, Vol. IV, R. J. Knops (ed.), 1979, 136-212. | MR | Zbl

[29] L. Tartar, Mathematical tools for studying oscillations and concentrations: from Young measures to H-measures and their variants, In: “Multiscale Problems in Science and Technology”, N. Antonić, C. J. van Duijn, W. Jäger and A. Mikelić (eds.), Springer, Berlin, 2002, 1-84. | MR | Zbl

[30] A. Visintin, Some properties of two-scale convergence, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15 (2004), 93-107. | EuDML | MR | Zbl

[31] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var. 12 (2006), 371-397. | EuDML | Numdam | MR | Zbl

[32] A. Visintin, Homogenization of doubly-nonlinear equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006), 211-222. | MR | Zbl

[33] A. Visintin, Two-scale convergence of first-order operators, Z. Anal. Anwendungen 26 (2007), 133-164. | MR | Zbl

[34] A. Visintin, Two-scale convergence of some integral functionals, Calc. Var. Partial Differential Equations 29 (2007), 239-265. | MR | Zbl