Geodesic
Some properties of two-scale convergence
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 2, pp. 93-107.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We reformulate and extend G. Nguetseng’s notion of two-scale convergence by means of a variable transformation, and outline some of its properties. We approximate two-scale derivatives, and extend this convergence to spaces of differentiable functions. The two-scale limit of derivatives of bounded sequences in the Sobolev spaces $W^{1,p}(\mathbb{R}^{N})$, $L^{2}_{rot}(\mathbb{R}^{3})^{3}$, $L^{2}_{div}(\mathbb{R}^{3})^{3}$ and $W^{2,p}(\mathbb{R}^{N})$ is then characterized. The two-scale limit behaviour of the potentials of a two-scale convergent sequence of irrotational fields is finally studied.
Mediante una trasformazione di variabile, la nozione di convergenza a due scale di G. Nguetseng è qui riformulata ed estesa, ed alcune delle sue proprietà sono presentate. Tale convergenza è quindi estesa a spazi di funzioni differenziabili mediante l’approssimazione delle derivate a due scale. Inoltre si caratterizza il limite a due scale di derivate di successioni limitate negli spazi di Sobolev $W^{1,p}(\mathbb{R}^{N})$, $L^{2}_{rot}(\mathbb{R}^{3})^{3}$, $L^{2}_{div}(\mathbb{R}^{3})^{3}$ e $W^{2,p}(\mathbb{R}^{N})$. Infine si studia il limite a due scale dei potenziali di una successione convergente a due scale di campi irrotazionali. 
@article{RLIN_2004_9_15_2_a3,
     author = {Visintin, Augusto},
     title = {Some properties of two-scale convergence},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {93--107},
     publisher = {mathdoc},
     volume = {Ser. 9, 15},
     number = {2},
     year = {2004},
     zbl = {1225.35031},
     mrnumber = {1185639},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_2_a3/}
}
TY  - JOUR
AU  - Visintin, Augusto
TI  - Some properties of two-scale convergence
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 2004
SP  - 93
EP  - 107
VL  - 15
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_2_a3/
LA  - en
ID  - RLIN_2004_9_15_2_a3
ER  - 
%0 Journal Article
%A Visintin, Augusto
%T Some properties of two-scale convergence
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2004
%P 93-107
%V 15
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_2_a3/
%G en
%F RLIN_2004_9_15_2_a3
Visintin, Augusto. Some properties of two-scale convergence. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 2, pp. 93-107. http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_2_a3/

[1] G. Allaire, Homogenization and two-scale convergence. S.I.A.M. J. Math. Anal., 23, 1992, 1482-1518. | DOI | 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. S.I.A.M. J. Math. Anal., 21, 1990, 823-836. | DOI | MR | Zbl

[4] A. Bensoussan - J.L. Lions - G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam 1978. | MR | Zbl

[5] A. Bourgeat - S. Luckhaus - A. Mikelic, Convergence of the homogenization process for a double-porosity model of immiscible two-phase flow. S.I.A.M. J. Math. Anal., 27, 1996, 1520-1543. | DOI | MR | Zbl

[6] J.K. Brooks - R.V. Chacon, Continuity and compactness of measures. Adv. in Math., 37, 1980, 16-26. | DOI | MR | Zbl

[7] J. Casado-Diaz - 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. | MR | Zbl

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

[9] D. Cioranescu - P. Donato, An Introduction to Homogenization. Oxford Univ. Press, New York 1999. | MR | Zbl

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

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

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

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

[14] F. Murat - L. Tartar, $H$-convergence. In: A. CHERKAEV - R. KOHN (eds.), Topics in the Mathematical Modelling of Composite Materials. Birkhäuser, Boston 1997, 21-44. | MR | Zbl

[15] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. S.I.A.M. J. Math. Anal., 20, 1989, 608-623. | DOI | MR | Zbl

[16] G. Nguetseng, Asymptotic analysis for a stiff variational problem arising in mechanics. S.I.A.M. J. Math. Anal., 21, 1990, 1394-1414. | DOI | MR | Zbl

[17] L. Tartar, Mathematical tools for studying oscillations and concentrations: from Young measures to $H$-measures and their variants. In: N. ANTONIĆ - C.J. VAN DUIJN - W. JÄGER - A. MIKELIĆ (eds.), Multiscale Problems in Science and Technology. Springer, Berlin 2002, 1-84. | MR | Zbl

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

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