Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-25T04:16:53.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards a two-scale calculus

Published online by Cambridge University Press:  20 June 2006

Augusto Visintin
Augusto Visintin*
Affiliation:
Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento), Italia; Visintin@science.unitn.it
Get access

Abstract

We define and characterize weak and strong two-scale convergence in Lp, C0 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.

Keywords

Two-scale convergencetwo-scale decompositionSobolev spaceshomogenization.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 3 , July 2006 , pp. 371 - 397
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. CrossRef
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.
G. Allaire, Shape Optimization by the Homogenization Method. Springer, New York (2002).
Allaire, G. and Briane, M., Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinburgh A 126 (1996) 297–342. CrossRef
Arbogast, T., Douglas, J. and Hornung, U., Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21 (1990) 823–836. CrossRef
Ball, J.M. and Murat, F., Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc. 107 (1989) 655–663.
G. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).
Bourgeat, A., Luckhaus, S. and Mikelić, A., Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow. SIAM J. Math. Anal. 27 (1996) 1520–1543. CrossRef
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998).
Brooks, J.K. and Chacon, R.V., Continuity and compactness of measures. Adv. Math. 37 (1980) 16–26. CrossRef
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.
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.
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.
A. Cherkaev, R. Kohn Eds., Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston (1997).
D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization. C.R. Acad. Sci. Paris, Ser. I 335 (2002) 99–104.
D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Univ. Press, New York (1999).
C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures. Wiley, Chichester and Masson, Paris (1995).
N. Dunford and J. Schwartz, Linear Operators. Vol. I. Interscience, New York (1958).
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer, Berlin.
M. Lenczner, Homogénéisation d'un circuit électrique. C.R. Acad. Sci. Paris, Ser. II 324 (1997) 537–542.
Lenczner, M. and Senouci, G., Homogenization of electrical networks including voltage-to-voltage amplifiers. Math. Models Meth. Appl. Sci. 9 (1999) 899–932. CrossRef
J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Springer, Berlin, 1972.
Lukkassen, D., Nguetseng, G. and Wall, P., Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002) 35–86.
F. Murat and L. Tartar, H-convergence. In [14], 21–44.
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. CrossRef
Nguetseng, G., Asymptotic analysis for a stiff variational problem arising in mechanics. SIAM J. Math. Anal. 21 (1990) 1394–1414. CrossRef
Nguetseng, G., Homogenization structures and applications, I. Zeit. Anal. Anwend. 22 (2003) 73–107. CrossRef
O.A. Oleĭnik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992).
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer, New York (1980).
L. Tartar, Course Peccot. Collège de France, Paris (1977). (Unpublished, partially written in [24]).
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.
Visintin, A., Vector Preisach model and Maxwell's equations. Physica B 306 (2001) 21–25. CrossRef
A. Visintin, Some properties of two-scale convergence. Rendic. Accad. Lincei XV (2004) 93–107.
A. Visintin, Two-scale convergence of first-order operators. (submitted)
Weinan, E., Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math. 45 (1992) 301–326. CrossRef
Zhikov, V.V., On an extension of the method of two-scale convergence and its applications. Sb. Math. 191 (2000) 973–1014. CrossRef