Towards a two-scale calculus

Visintin, Augusto

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 371-397.

We define and characterize weak and strong two-scale convergence in $L^p$, $C^0$ and other spaces via a transformation of variable, extending Nguetseng’s definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.

Classification : 35B27, 35J20, 74Q, 78M40
Mots clés : two-scale convergence, two-scale decomposition, Sobolev spaces, homogenization

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

[2] G. Allaire, Homogenization of the unsteady Stokes equations in porous media, in Progress in Partial Differential Equations: Calculus of Variations, Applications, C. Bandle Ed. Longman, Harlow (1992) 109-123. | Zbl

[3] G. Allaire, Shape Optimization by the Homogenization Method. Springer, New York (2002). | MR | Zbl

[4] G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinburgh A 126 (1996) 297-342. | Zbl

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

[6] J.M. Ball and F. Murat, Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc. 107 (1989) 655-663. | Zbl

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

[8] 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. | Zbl

[9] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). | MR | Zbl

[10] J.K. Brooks and R.V. Chacon, Continuity and compactness of measures. Adv. Math. 37 (1980) 16-26. | Zbl

[11] J. Casado-Diaz and I. Gayte, A general compactness result and its application to two-scale convergence of almost periodic functions. C. R. Acad. Sci. Paris, Ser. I 323 (1996) 329-334. | Zbl

[12] J. Casado-Diaz and I. Gayte, The two-scale convergence method applied to generalized Besicovitch spaces. R. Soc. Lond. Proc., Ser. A 458 (2002) 2925-2946. | Zbl

[13] J. Casado-Diaz, M. Luna-Laynez and J.D. Martin, An adaptation of the multi-scale method for the analysis of very thin reticulated structures. C. R. Acad. Sci. Paris, Ser. I 332 (2001) 223-228. | Zbl

[14] A. Cherkaev, R. Kohn Eds., Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston (1997). | MR | Zbl

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

[16] D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Univ. Press, New York (1999). | MR | Zbl

[17] C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures. Wiley, Chichester and Masson, Paris (1995). | MR | Zbl

[18] N. Dunford and J. Schwartz, Linear Operators. Vol. I. Interscience, New York (1958). | Zbl

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

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

[21] M. Lenczner and G. Senouci, Homogenization of electrical networks including voltage-to-voltage amplifiers. Math. Models Meth. Appl. Sci. 9 (1999) 899-932. | Zbl

[22] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Springer, Berlin, 1972. | MR | Zbl

[23] D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002) 35-86. | Zbl

[24] F. Murat and L. Tartar, H-convergence. In [14], 21-44.

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

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

[27] G. Nguetseng, Homogenization structures and applications, I. Zeit. Anal. Anwend. 22 (2003) 73-107. | Zbl

[28] O.A. Oleĭnik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992). | MR | Zbl

[29] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer, New York (1980). | Zbl

[30] L. Tartar, Course Peccot. Collège de France, Paris (1977). (Unpublished, partially written in [24]).

[31] 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, A. Mikelić Eds. Springer, Berlin (2002) 1-84. | Zbl

[32] A. Visintin, Vector Preisach model and Maxwell's equations. Physica B 306 (2001) 21-25.

[33] A. Visintin, Some properties of two-scale convergence. Rendic. Accad. Lincei XV (2004) 93-107.

[34] A. Visintin, Two-scale convergence of first-order operators. (submitted) | Zbl

[35] E. Weinan, Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math. 45 (1992) 301-326. | Zbl

[36] V.V. Zhikov, On an extension of the method of two-scale convergence and its applications. Sb. Math. 191 (2000) 973-1014. | Zbl