Skip to main content
SpringerLink
Log in

Analysis of the asymptotic distance between oscillating functions and their weak limit in L 2

  • Chapter
Advances in Mathematical Economics

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 1))

  • 433 Accesses

Summary

Young measures theory is applied to the understanding of weak convergence without strong convergence in L 2 spaces. The two-scale Young measures permit also to analyse, when it happens, a “modulated periodical” behavior and, in the general case, to get a kind of orthogonal decomposition. Some new examples are discussed.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alexandre, R.: Homogénéisation et θ — 2 convergence. C.R. Acad. Sci. Paris 320, 787–790 (1995)

    Google Scholar 

  2. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)

    Article  Google Scholar 

  3. Allaire, G.: Two-scale convergence and homogenization of periodic structures. In: School on homogenization. Lecture notes of the courses held at ICTP, Trieste, September 6–17, 1993. SISSA Ref. 140/93/M, Trieste, 1993. pp.1–23

    Google Scholar 

  4. Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenisation. Proc. Royal Soc. Edinburgh 126A, 297–342 (1996)

    Article  Google Scholar 

  5. Amrani, A., Castaing, C., Valadier, M.: Méthodes de troncature appliquées à des problèmes de convergence faible ou forte dans L 1 . Arch. Rational Mech. Anal. 117, 167–191 (1992)

    Article  Google Scholar 

  6. Balder, E.J.: A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22, 570–598 (1984)

    Article  Google Scholar 

  7. Balder, E.J.: On weak convergence implying strong convergence in L 1-spaces. Bull. Austral. Math. Soc. 33, 363–368 (1986)

    Article  Google Scholar 

  8. Balder, E.J.: Generalized equilibrium results for games with incomplete information. Math. Oper. Res. 13–2, 265–276 (1988)

    Article  Google Scholar 

  9. Balder, E.J.: Lectures on Young measures. Preprint n° 9517, CEREMADE, Université Paris-Dauphine, Prance 1995

    Google Scholar 

  10. Balder, E.J.: On compactness results for multi-scale convergence. University of Utrecht 1995 (not yet published)

    Google Scholar 

  11. Balder, E.J., Girardi M., Jalby, V.: Prom weak to strong types of ℒ l E -convergence by the Bocce-criterion. Studia Math. 111, 241–262 (1994)

    Google Scholar 

  12. Ball, J., Murat, F.: W 1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58, 225–253 (1984)

    Article  Google Scholar 

  13. Bourgeat, A., Mikelic, A., Wright, S.: Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math. 456, 19–51 (1994)

    Google Scholar 

  14. Castaing, C.: Convergence faible et sections extrémales. Sém. Anal. Convexe de Montpellier 18, 2.1–2.18 (1988)

    Google Scholar 

  15. Dacorogna, B.: Direct Methods in The Calculus of Variations. Springer-Verlag 1989

    Book  Google Scholar 

  16. Dudley, R.M.: Real analysis and probability. Wadsworth & Brooks/Cole Mathematics Series. California 1989

    Google Scholar 

  17. E., W.: Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math. 45, 301–326 (1992)

    Google Scholar 

  18. Ekeland, I., Temam, R.: Analyse convexe et problèmes variationnels. Dunod Gauthier-Villars, Paris 1974 (English edition: North-Holland, Amsterdam, 1976)

    Google Scholar 

  19. Halmos, P.R.: Measure Theory (tenth edition). Van Nostrand, Princeton 1965

    Google Scholar 

  20. Komlös, J.: A generalisation of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217–229 (1967)

    Article  Google Scholar 

  21. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989)

    Article  Google Scholar 

  22. Roubíček, T: Relaxation in optimization theory and variational calculus. W. de Gruyter, Berlin 1997

    Book  Google Scholar 

  23. Rzuzuchowski, T.: Strong convergence of selections implied by weak. Bull. Austral. Math. Soc. 39, 201–214 (1989)

    Article  Google Scholar 

  24. Saadoune, M., Valadier M.: Convergence in measure. The Fréchet criterion from local to global. Bull. Polish Acad. Sci. Math. 43, 47–57 (1995)

    Google Scholar 

  25. Saadoune, M., Valadier, M.: Extraction of a “good” subsequence from a bounded sequence of integrable functions. J. Convex Anal. 2, 345–357 (1995)

    Google Scholar 

  26. Scorza Dragoni, G.: Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un’altra variabile. Rend. Sem. Mat. Univ. Padova 17, 102–106 (1948)

    Google Scholar 

  27. Suquet, P.: Plasticité et homogénéisation. Thèse de Doctorat d’État, Paris 1982

    Google Scholar 

  28. Tartar, L.: Une nouvelle méthode de résolution d’équations aux dérivées partielles non linéaires. In: Journées d’Analyse Non Linéaire (P. Bénilan, J. Robert eds.). Lecture Notes in Math. 665, pp.228–241 Springer-Verlag 1978

    Chapter  Google Scholar 

  29. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. IV (R.J. Knops ed.). Research Notes in Math. 39, pp.136–212 Pitman, London 1979

    Google Scholar 

  30. Valadier, M.: Différents cas où, grâce à une propriété d’extrémalité, une suite de fonctions intégrables faiblement convergente converge fortement. Sém. Anal. Convexe 19, 5.1–5.20 (1989)

    Google Scholar 

  31. Valadier, M.: Young measures. In: Methods of Nonconvex Analysis (A. Cellina ed.). Lecture Notes in Math. 1446, pp.152–188 Springer-Verlag 1990

    Chapter  Google Scholar 

  32. Valadier, M.: A course on Young measures. Workshop di Teoria delia Misura e Analisi Reale, Grado, September 19-October 2, 1993. Rend. Istit. Mat. Univ. Trieste 26, suppl., 349–394 (1994)

    Google Scholar 

  33. Valadier, M.: Two-scale convergence and Young measures. Prépublication n° 02 (15 pages). Départ, de Math., Université de Montpellier, France 1995

    Google Scholar 

  34. Valadier, M.: Admissible functions in two-scale convergence. Portugaliae Math. 54, 147–164 (1997)

    Google Scholar 

  35. Visintin, A.: Strong convergence results related to strict convexity. Comm. Partial Differential Equations 9, 439–466 (1984)

    Article  Google Scholar 

  36. Young, L.C.: Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations. Comptes Rendus de la Société des Sc. et des lettres de Varsovie cl. III 30, 212–234 (1937)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques - case 051, Université Montpellier II, place Eugène Bataillon, 34 095, Montpellier Cedex 5, France

    Michel Valadier

Authors
  1. Michel Valadier
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-0041, Japan

    Shigeo Kusuoka (Professor) (Professor)

  2. Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan

    Toru Maruyama (Professor) (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag

About this chapter

Cite this chapter

Valadier, M. (1999). Analysis of the asymptotic distance between oscillating functions and their weak limit in L 2 . In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-65895-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-65895-5_7

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-65897-9

  • Online ISBN: 978-4-431-65895-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation