Skip to main content
SpringerLink
Log in

Variational formulation and structural stability of monotone equations

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

After Fitzpatrick’s seminal work [MR 1009594], it is known that in a real Banach space V any maximal monotone operator \({\alpha: V \to {\mathcal P}(V^{\prime})}\) may be given a variational representation. This is here illustrated on some examples. On this basis, De Giorgi’s notion of Γ-convergence is then applied to the analysis of monotone inclusions, like \({D_tu + \alpha(u)\ni h}\) . The compactness and the structural stability are studied, with respect to variations of the operator α and of the datum h. The possible onset of long memory in the limit is also discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosetti A., Sbordone C.: Γ-convergenza e G-convergenza per problemi non lineari di tipo ellittico. Boll. Un. Mat. Ital 13A(5), 352–362 (1976)

    MathSciNet  Google Scholar 

  2. Attouch H.: Variational Convergence for Functions and Operators. Pitman, Boston (1984)

    MATH  Google Scholar 

  3. Auchmuty G.: Saddle-points and existence-uniqueness for evolution equations. Differ. Integral Equ. 6, 1161–1171 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Barbu V.: Existence theorems for a class of two point boundary problems. J. Differ. Equ. 17, 236–257 (1975)

    Article  MathSciNet  Google Scholar 

  5. Barbu V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  6. Barbu V., Precupanu T.: Convexity and Optimization in Banach Spaces. Editura Academiei, Bucuresti (1978)

    Google Scholar 

  7. Braides A.: Γ-Convergence for Beginners. Oxford University Press, Oxford (2002)

    Book  MATH  Google Scholar 

  8. Braides A., Defranceschi A.: Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  9. Brezis H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  10. Brezis H.: Analyse Fonctionelle. Théorie et Applications. Masson, Paris (1983)

    Google Scholar 

  11. 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

  12. Browder, F.: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math., XVIII Part II. A.M.S., Providence (1976)

  13. Buliga M., de Saxcé G., Vallée C.: Existence and construction of bipotentials for graphs of multivalued laws. J. Convex. Anal. 15, 87–104 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Buliga M., de Saxcé G., Vallée C.: Bipotentials for non-monotone multivalued operators: fundamental results and applications. Acta Appl. Math. 110, 955–972 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Burachik R.S., Svaiter B.F.: Maximal monotone operators, convex functions, and a special family of enlargements. Set Valued Anal. 10, 297–316 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Burachik R.S., Svaiter B.F.: Maximal monotonicity, conjugation and the duality product. Proc. Am. Math. Soc 131, 2379–2383 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Checcucci V., Tognoli A., Vesentini E.: Lezioni di Topologia Generale. Feltrinelli, Milano (1972)

    Google Scholar 

  18. Dal Maso G.: An Introduction to Γ-Convergence. Birkhäuser, Boston (1993)

    Book  Google Scholar 

  19. De Giorgi E., Franzoni T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 58(8), 842–850 (1975)

    MathSciNet  MATH  Google Scholar 

  20. De Giorgi E., Spagnolo S.: Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital 8, 391–411 (1973)

    MathSciNet  MATH  Google Scholar 

  21. Dontchev A.L., Zolezzi T.: Well-Posed Optimization Problems. Springer, Berlin (1993)

    MATH  Google Scholar 

  22. Ekeland I., Temam R.: Analyse Convexe et Problèmes Variationnelles. Dunod Gauthier-Villars, Paris (1974)

    Google Scholar 

  23. Fenchel W.: Convex Cones, Sets, and Functions. Princeton University, Princeton (1953)

    MATH  Google Scholar 

  24. 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)

  25. Ghoussoub N.: A variational theory for monotone vector fields. J. Fixed Point Theory Appl. 4, 107–135 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ghoussoub N.: Self-Dual Partial Differential Systems and their Variational Principles. Springer, Berlin (2009)

    MATH  Google Scholar 

  27. Ghoussoub N., Tzou L: A variational principle for gradient flows. Math. Ann. 330, 519–549 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hiriart-Urruty J.-B., Lemarechal C.: Convex Analysis and Optimization Algorithms. Springer, Berlin (1993)

    Google Scholar 

  29. Marques Alves M., Svaiter B.F.: Brøndsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces. J. Convex Anal. 15, 693–706 (2008)

    MathSciNet  MATH  Google Scholar 

  30. Martinez-Legaz J.-E., Svaiter B.F.: Monotone operators representable by l.s.c. convex functions. Set Valued Anal. 13, 21–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Martinez-Legaz J.-E., Svaiter B.F.: Minimal convex functions bounded below by the duality product. Proc. Am. Math. Soc 136, 873–878 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Martinez-Legaz J.-E., Théra M.: A convex representation of maximal monotone operators. J. Nonlinear Convex Anal. 2, 243–247 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Nayroles B.: Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B 282, A1035–A1038 (1976)

    MathSciNet  Google Scholar 

  34. Penot J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58, 855–871 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rockafellar R.T.: A general correspondence between dual minimax problems and convex programs. Pacific J. Math. 25, 597–611 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1969)

    Google Scholar 

  37. Stefanelli U.: The Brezis-Ekeland principle for doubly nonlinear equations. SIAM J. Control Optim 8, 1615–1642 (2008)

    Article  MathSciNet  Google Scholar 

  38. Svaiter B.F.: Fixed points in the family of convex representations of a maximal monotone operator. Proc. Am. Math. Soc 131, 3851–3859 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tartar, L.: Nonlocal effects induced by homogenization. In: Colombini, F., Marino, A., Modica, L., Spagnolo, S. (eds.) Partial Differential Equations and the Calculus of Variations, vol. II, pp. 925–938. Birkhäuser, Boston (1989)

  40. Tartar L.: Memory effects and homogenization. Arch. Rational Mech. Anal. 111, 121–133 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tartar, L.: The General Theory of Homogenization. A Personalized Introduction. Springer, Berlin; UMI, Bologna, (2009)

  42. Visintin A.: Extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl. 18, 633–650 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Visintin A.: Scale-transformations in the homogenization of nonlinear magnetic processes. Arch. Rat. Mech. Anal. 198, 569–611 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Visintin A.: Homogenization of a parabolic model of ferromagnetism. J. Differ. Eq. 250, 1521–1552 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Visintin A.: Scale-transformations and homogenization of maximal monotone relations, and applications. Arch. Rational Mech. Anal. 198(2), 569–611 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Visintin A.: Structural stability of doubly nonlinear flows. Boll. Un. Mat. Ital IV, 363–391 (2011)

    MathSciNet  Google Scholar 

  47. Visintin A.: On the structural stability of monotone flows. Boll. Un. Mat. Ital IV, 471–479 (2011)

    MathSciNet  Google Scholar 

  48. Visintin, A.: Structural stability of rate-independent nonpotential flows. (in press)

  49. Visintin, A.: Compactness and structural stability of pseudo-monotone equations. (in progress)

  50. Zeidler E.: Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators. Springer, New York (1990)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, Povo, 38050, Trento, Italy

    Augusto Visintin

Authors
  1. Augusto Visintin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Augusto Visintin.

Additional information

Communicated by L. Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Visintin, A. Variational formulation and structural stability of monotone equations. Calc. Var. 47, 273–317 (2013). https://doi.org/10.1007/s00526-012-0519-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-012-0519-y

Mathematics Subject Classification

Search

Navigation