Towards a two-scale calculus

Augusto Visintin

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

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

Abstract

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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.

How to cite

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Visintin, 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

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  1. G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518.  
  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.  
  3. G. Allaire, Shape Optimization by the Homogenization Method. Springer, New York (2002).  
  4. G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinburgh A126 (1996) 297–342.  
  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.  
  6. J.M. Ball and F. Murat, Remarks on Chacon's biting lemma. Proc. Amer. Math. Soc.107 (1989) 655–663.  
  7. G. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).  
  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.  
  9. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998).  
  10. J.K. Brooks and R.V. Chacon, Continuity and compactness of measures. Adv. Math.37 (1980) 16–26.  
  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. I323 (1996) 329–334.  
  12. 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.  
  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. I332 (2001) 223–228.  
  14. A. Cherkaev, R. Kohn Eds., Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston (1997).  
  15. D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization. C.R. Acad. Sci. Paris, Ser. I335 (2002) 99–104.  
  16. D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Univ. Press, New York (1999).  
  17. C. Conca, J. Planchard and M. Vanninathan, Fluids and Periodic Structures. Wiley, Chichester and Masson, Paris (1995).  
  18. N. Dunford and J. Schwartz, Linear Operators. Vol. I. Interscience, New York (1958).  
  19. V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer, Berlin.  
  20. M. Lenczner, Homogénéisation d'un circuit électrique. C.R. Acad. Sci. Paris, Ser. II324 (1997) 537–542.  
  21. M. Lenczner and G. Senouci, Homogenization of electrical networks including voltage-to-voltage amplifiers. Math. Models Meth. Appl. Sci.9 (1999) 899–932.  
  22. J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Springer, Berlin, 1972.  
  23. D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence. Int. J. Pure Appl. Math.2 (2002) 35–86.  
  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.  
  26. G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics. SIAM J. Math. Anal.21 (1990) 1394–1414.  
  27. G. Nguetseng, Homogenization structures and applications, I. Zeit. Anal. Anwend.22 (2003) 73–107.  
  28. O.A. Oleĭnik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992).  
  29. E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer, New York (1980).  
  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.  
  32. A. Visintin, Vector Preisach model and Maxwell's equations. Physica B306 (2001) 21–25.  
  33. A. Visintin, Some properties of two-scale convergence. Rendic. Accad. LinceiXV (2004) 93–107.  
  34. A. Visintin, Two-scale convergence of first-order operators. (submitted)  
  35. E. Weinan, Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math.45 (1992) 301–326.  
  36. V.V. Zhikov, On an extension of the method of two-scale convergence and its applications. Sb. Math.191 (2000) 973–1014.  

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