# Towards a two-scale calculus

ESAIM: Control, Optimisation and Calculus of Variations (2006)

- Volume: 12, Issue: 3, page 371-397
- ISSN: 1292-8119

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topVisintin, Augusto. "Towards a two-scale calculus." ESAIM: Control, Optimisation and Calculus of Variations 12.3 (2006): 371-397. <http://eudml.org/doc/249669>.

@article{Visintin2006,

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.
},

author = {Visintin, Augusto},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Two-scale convergence;
two-scale decomposition;
Sobolev spaces;
homogenization.; two-scale convergence; two-scale decomposition; two-scale convolution},

language = {eng},

month = {6},

number = {3},

pages = {371-397},

publisher = {EDP Sciences},

title = {Towards a two-scale calculus},

url = {http://eudml.org/doc/249669},

volume = {12},

year = {2006},

}

TY - JOUR

AU - Visintin, Augusto

TI - Towards a two-scale calculus

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2006/6//

PB - EDP Sciences

VL - 12

IS - 3

SP - 371

EP - 397

AB -
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.

LA - eng

KW - Two-scale convergence;
two-scale decomposition;
Sobolev spaces;
homogenization.; two-scale convergence; two-scale decomposition; two-scale convolution

UR - http://eudml.org/doc/249669

ER -

## References

top- G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518. Zbl0770.35005
- 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. Zbl0801.35103
- G. Allaire, Shape Optimization by the Homogenization Method. Springer, New York (2002). Zbl0990.35001
- G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinburgh A126 (1996) 297–342. Zbl0866.35017
- 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. Zbl0698.76106
- J.M. Ball and F. Murat, Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc.107 (1989) 655–663. Zbl0678.46023
- G. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978). Zbl0404.35001
- 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. Zbl0866.35018
- A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). Zbl0911.49010
- J.K. Brooks and R.V. Chacon, Continuity and compactness of measures. Adv. Math.37 (1980) 16–26. Zbl0463.28003
- 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. I323 (1996) 329–334. Zbl0865.46003
- J. Casado-Diaz and I. Gayte, The two-scale convergence method applied to generalized Besicovitch spaces. R. Soc. Lond. Proc., Ser. A458 (2002) 2925–2946. Zbl1099.35010
- 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. I332 (2001) 223–228. Zbl0984.35017
- 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. I335 (2002) 99–104. Zbl1001.49016
- D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Univ. Press, New York (1999). Zbl0939.35001
- C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures. Wiley, Chichester and Masson, Paris (1995). Zbl0910.76002
- N. Dunford and J. Schwartz, Linear Operators. Vol. I. Interscience, New York (1958). Zbl0084.10402
- 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. II324 (1997) 537–542. Zbl0887.35016
- M. Lenczner and G. Senouci, Homogenization of electrical networks including voltage-to-voltage amplifiers. Math. Models Meth. Appl. Sci.9 (1999) 899–932. Zbl0963.35014
- J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Springer, Berlin, 1972. Zbl0223.35039
- D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence. Int. J. Pure Appl. Math.2 (2002) 35–86. Zbl1061.35015
- F. Murat and L. Tartar, H-convergence. In [14], 21–44.
- G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608–623. Zbl0688.35007
- G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics. SIAM J. Math. Anal.21 (1990) 1394–1414. Zbl0723.73011
- G. Nguetseng, Homogenization structures and applications, I. Zeit. Anal. Anwend.22 (2003) 73–107. Zbl1045.46031
- O.A. Oleĭnik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992). Zbl0768.73003
- E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer, New York (1980). Zbl0432.70002
- 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. Zbl1015.35001
- A. Visintin, Vector Preisach model and Maxwell's equations. Physica B306 (2001) 21–25.
- A. Visintin, Some properties of two-scale convergence. Rendic. Accad. LinceiXV (2004) 93–107. Zbl1225.35031
- A. Visintin, Two-scale convergence of first-order operators. (submitted) Zbl1128.35018
- E. Weinan, Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math.45 (1992) 301–326. Zbl0794.35014
- V.V. Zhikov, On an extension of the method of two-scale convergence and its applications. Sb. Math.191 (2000) 973–1014. Zbl0969.35048

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