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

Augusto Visintin

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 3, page 471-479
  • ISSN: 0392-4041

Abstract

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Flows of the form D t ⁢ u + α ⁢ ( u ) ∋ h , with α maximal monotone, are here formulated as null-minimization problems via Fitzpatrick's theory. By means of De Giorgi's notion of Γ -convergence, we study the compactness and the structural stability of these flows with respect to variations of the source h and of the operator α .

How to cite

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Visintin, Augusto. "On the Structural Stability of Monotone Flows (Running head: Structural Stability)." Bollettino dell'Unione Matematica Italiana 4.3 (2011): 471-479. <http://eudml.org/doc/290730>.

@article{Visintin2011,
abstract = {Flows of the form $D_tu + \alpha(u) \ni h$, 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 $\alpha$.},
author = {Visintin, Augusto},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {471-479},
publisher = {Unione Matematica Italiana},
title = {On the Structural Stability of Monotone Flows (Running head: Structural Stability)},
url = {http://eudml.org/doc/290730},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Visintin, Augusto
TI - On the Structural Stability of Monotone Flows (Running head: Structural Stability)
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/10//
PB - Unione Matematica Italiana
VL - 4
IS - 3
SP - 471
EP - 479
AB - Flows of the form $D_tu + \alpha(u) \ni h$, 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 $\alpha$.
LA - eng
UR - http://eudml.org/doc/290730
ER -

References

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  2. ATTOUCH, H., Variational Convergence for Functions and Operators. Pitman, Boston1984 Zbl0561.49012MR773850
  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. Zbl0332.49032MR637214
  4. DAL MASO, G., An Introduction to Γ -Convergence. Birkhäuser, Boston1993. Zbl0816.49001MR1201152DOI10.1007/978-1-4612-0327-8
  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. MR448194
  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. Zbl0669.47029MR1009594
  7. GHOUSSOUB, N., A variational theory for monotone vector fields. J. Fixed Point Theory Appl., 4 (2008), 107-135. Zbl1177.35093MR2447965DOI10.1007/s11784-008-0083-4
  8. GHOUSSOUB, N., Selfdual Partial Differential Systems and their Variational Principles. Springer, 2009. Zbl1357.49004MR2458698
  9. NAYROLES, B., Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B, 282 (1976). A1035-A1038. Zbl0345.73037MR418609
  10. STEFANELLI, U., The Brezis-Ekeland principle for doubly nonlinear equations. S.I.A.M. J. Control Optim., 8 (2008), 1615-1642. Zbl1194.35214MR2425653DOI10.1137/070684574
  11. TARTAR, L., The General Theory of Homogenization. A Personalized Introduction. Springer, Berlin; UMI, Bologna, 2009. Zbl1188.35004MR2582099DOI10.1007/978-3-642-05195-1
  12. VISINTIN, A., Extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl., 18 (2008), 633-650. Zbl1191.47067MR2489147
  13. VISINTIN, A., Scale-transformations of maximal monotone relations in view of homogenization. Boll. Un. Mat. Ital., (9), III (2010), 591-601. MR2742783
  14. VISINTIN, A., Homogenization of a parabolic model of ferromagnetism. J. Differential Equations, 250 (2011), 1521-1552. Zbl1213.35066MR2737216DOI10.1016/j.jde.2010.09.016
  15. VISINTIN, A., Structural stability of doubly nonlinear flows. Boll. Un. Mat. Ital., (9), IV (2011) (in press). Zbl1235.35032MR2906767
  16. VISINTIN, A., Variational formulation and structural stability of monotone equations. (forthcoming). Zbl1304.47073MR3044140DOI10.1007/s00526-012-0519-y

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