Scale-Transformations of Maximal Monotone Relations in View of Homogenization
Visintin, Augusto
Bollettino dell'Unione Matematica Italiana, Tome 3 (2010), p. 591-601 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

On the basis of Fitzpatrick's variational formulation of maximal monotone relations, and of Nguetseng's two-scale approach to homogenization, scale-transformations have recently been introduced and used for the periodic homogenization of quasilinear P.D.E.s. This note illustrates some basic results of this method.

Publié le : 2010-10-01
@article{BUMI_2010_9_3_3_591_0,
     author = {Augusto Visintin},
     title = {Scale-Transformations of Maximal Monotone Relations in View of Homogenization},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {3},
     year = {2010},
     pages = {591-601},
     mrnumber = {2742783},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2010_9_3_3_591_0}
}

Visintin, Augusto. Scale-Transformations of Maximal Monotone Relations in View of Homogenization. Bollettino dell'Unione Matematica Italiana, Tome 3 (2010) pp. 591-601. http://gdmltest.u-ga.fr/item/BUMI_2010_9_3_3_591_0/

[1] Allaire, G., Homogenization and two-scale convergence. S.I.A.M. J. Math. Anal., 23 (1992), 1482-1518. | MR 1185639 | Zbl 0770.35005

[2] Barbu, V., Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, Berlin2010. | MR 2582280 | Zbl 1197.35002

[3] Brezis, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam1973. | MR 348562 | Zbl 0252.47055

[4] Brezis, H. - Ekeland, I., Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps, and II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), 971-974. | MR 637214 | Zbl 0332.49032

[5] Dal Maso, G., An Introduction to Γ-Convergence. Birkhäuser, Boston1993. | MR 1201152 | Zbl 0816.49001

[6] De Giorgi, E. - Franzoni, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 58 (8) (1975), 842-850. | MR 448194

[7] Ekeland, I. - Temam, R., Analyse Convexe et Problèmes Variationnelles. Dunod Gauthier-Villars, Paris 1974. | MR 463993

[8] Fitzpatrick, S., Representing monotone operators by convex functions. Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59-65; Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988. | MR 1009594 | Zbl 0669.47029

[9] Marcellini, P., Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl., 117 (1978), 139-152. | MR 515958 | Zbl 0395.49007

[10] Martinez-Legaz, J.-E. - Svaiter, B. F., Monotone operators representable by l.s.c. convex functions. Set-Valued Anal., 13 (2005), 21-46. | MR 2128696 | Zbl 1083.47036

[11] Nayroles, B., Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1035-A1038. | MR 418609 | Zbl 0345.73037

[12] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. S.I.A.M. J. Math. Anal., 20 (1989), 608-623. | MR 990867 | Zbl 0688.35007

[13] Rockafellar, R. T., Convex Analysis. Princeton University Press, Princeton1969. | MR 274683

[14] Visintin, A., Homogenization of the nonlinear Kelvin-Voigt model of viscoelasticity and of the Prager model of plasticity. Continuum Mech. Thermodyn., 18 (2006), 223-252. | MR 2245987 | Zbl 1160.74331

[15] Visintin, A., Homogenization of the nonlinear Maxwell model of visco-elasticity and of the Prandtl-Reuss model of elasto-plasticity. Royal Soc. Edinburgh Proc. A, 138 (2008), 1-39. | MR 2488064

[16] Visintin, A., Homogenization of nonlinear visco-elastic composites. J. Math. Pures Appl., 89 (2008), 477-504. | MR 2416672 | Zbl 1166.35004

[17] Visintin, A., Extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl., 18 (2008), 633-650. | MR 2489147 | Zbl 1191.47067

[18] Visintin, A., Scale-integration and scale-disintegration in nonlinear homogenization. Calc. Var. Partial Differential Equations, 36 (2009), 565-590. | MR 2558331 | Zbl 1184.35041

[19] Visintin, A., Scale-transformations in the homogenization of nonlinear magnetic processes. Archive Rat. Mech. Anal. (in press). | MR 2721590 | Zbl 1233.78043

[20] Visintin, A., Homogenization of processes in nonlinear visco-elastic composites. Ann. Scuola Norm. Sup. Pisa (in press). | MR 2905380 | Zbl 1242.35033

[21] Visintin, A., A minimization principle for monotone equations. (submitted).

[22] Visintin, A., Scale-transformations and homogenization of maximal monotone relations, with applications. (forthcoming). | MR 3086566 | Zbl 1302.35042

[23] Visintin, A., Homogenization of a parabolic model of ferromagnetism. (forthcoming). | MR 2737216 | Zbl 1213.35066

[24] Zeidler, E., Nonlinear Functional Analysis and its Applications. Vol. II/B: Nonlinear Monotone Operators. Springer, New York1990. | MR 1033498 | Zbl 0684.47029