On the Structural Stability of Monotone Flows (Running head: Structural Stability)

Visintin, Augusto

Bollettino dell'Unione Matematica Italiana, Tome 4 (2011), p. 471-479 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Résumé

Flows of the form $D_{t}u+\alpha(u)\ni h$ , with $\alpha$ maximal monotone, are here formulated as null-minimization problems via Fitzpatrick's theory. By means of De Giorgi's notion of $\Gamma$ -convergence, we study the compactness and the structural stability of these flows with respect to variations of the source $h$ and of the operator $\alpha$ .

Publié le : 2011-10-01

MR 2906771 | Zbl 1243.49015

@article{BUMI_2011_9_4_3_471_0,
     author = {Augusto Visintin},
     title = {On the Structural Stability of Monotone Flows (Running head: Structural Stability)},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {4},
     year = {2011},
     pages = {471-479},
     zbl = {1243.49015},
     mrnumber = {2906771},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2011_9_4_3_471_0}
}

Visintin, Augusto. On the Structural Stability of Monotone Flows (Running head: Structural Stability). Bollettino dell'Unione Matematica Italiana, Tome 4 (2011) pp. 471-479. http://gdmltest.u-ga.fr/item/BUMI_2011_9_4_3_471_0/

Bibliographie

[1] Ambrosetti, A. - Sbordone, C., $\Gamma$ -convergenza e G-convergenza per problemi non lineari di tipo ellittico. Boll. Un. Mat. Ital. (5), 13-A (1976), 352-362 | MR 487703 | Zbl 0345.49004

[2] Attouch, H., Variational Convergence for Functions and Operators. Pitman, Boston1984 | MR 773850 | Zbl 0561.49012

[3] 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, and ibid. 1197-1198. | MR 637214 | Zbl 0332.49032

[4] Dal Maso, G., An Introduction to $\Gamma$ -Convergence. Birkhäuser, Boston1993. | MR 1201152 | Zbl 0816.49001

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

[6] 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

[7] Ghoussoub, N., A variational theory for monotone vector fields. J. Fixed Point Theory Appl., 4 (2008), 107-135. | MR 2447965 | Zbl 1177.35093

[8] Ghoussoub, N., Selfdual Partial Differential Systems and their Variational Principles. Springer, 2009. | MR 2458698 | Zbl 1357.49004

[9] 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

[10] Stefanelli, U., The Brezis-Ekeland principle for doubly nonlinear equations. S.I.A.M. J. Control Optim., 8 (2008), 1615-1642. | MR 2425653 | Zbl 1194.35214

[11] Tartar, L., The General Theory of Homogenization. A Personalized Introduction. Springer, Berlin; UMI, Bologna, 2009. | MR 2582099 | Zbl 1188.35004

[12] 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

[13] Visintin, A., Scale-transformations of maximal monotone relations in view of homogenization. Boll. Un. Mat. Ital., (9), III (2010), 591-601. | MR 2742783

[14] Visintin, A., Homogenization of a parabolic model of ferromagnetism. J. Differential Equations, 250 (2011), 1521-1552. | MR 2737216 | Zbl 1213.35066

[15] Visintin, A., Structural stability of doubly nonlinear flows. Boll. Un. Mat. Ital., (9), IV (2011) (in press). | MR 2906767 | Zbl 1235.35032

[16] Visintin, A., Variational formulation and structural stability of monotone equations. (forthcoming). | MR 3044140 | Zbl 1304.47073