Abstract

Given a σ\sigma-finite measure space (Ω,T,μ)(\Omega, \mathcal{T}, \mu) endowed with a μ\mu-complete tribe, a separable Banach space EE, we consider a topological vector space (X,T)(X,T), XX being a decomposable subspace of measurable EE-valued functions defined on Ω\Omega. Under a reasonable assumption on the vector topology TT, we show that if (fn)n{(f_{n})}_{n} is a sequence of extended real-valued measurable integrands defined on the product Ω×E\Omega\times E, with upper epi-limit (or upper Γ\Gamma-limit) f=lsefnf=ls_{e} f_{n}, then IfI_f is in many cases an upper bound for the TT-upper epi-limit of the sequence (Ifn)n(I_{f_{n}})_n, where IfI_f, IfnI_{f_{n}} are the integral functionals defined on XX associated to the integrands ff, fnf_n. The cases of Lebesgue spaces endowed with its strong, weak, or Mackey topologies are reached. We discuss also the necessity of the given conditions.

Author information

Contact details are reproduced from the original publication and may be historical.

Emmanuel Giner

Institut de Mathématiques, Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse 04, France

giner@math.univ-toulouse.fr

Keywords

  • Integral functional
  • upper epi-limits
  • Γ-convergence
  • finally equi-integrable sets

Mathematics Subject Classification

26A16, 26A24, 26E15, 28B20, 49J52, 54C35

References

  1. H. Attouch: Variational Convergence for Functions and Operators, Pitman, Boston (1984).Google Scholar
  2. J. P. Aubin, H. Frankowska: Set-Valued Analysis, Systems & Control: Foundations & Applications 2, Birkhäuser, Boston (1990).Google Scholar
  3. G. Beer: The slice topology: a viable alternative to Mosco convergence in nonreflexive spaces, Semin. d’Anal. Convexe, Montpellier 21, Exposé 3 (1991), 35 p.https://doi.org/10.1016/0362-546X(92)90145-5Google Scholar
  4. A. Bourass: Comparaison de Fonctionnelles Intégrales. Conditions de Croissance Caractéristiques et Application à la Semi Continuité dans les Espaces Intégraux de Type Orlicz, Thèse d’État, Perpignan (1983).Google Scholar
  5. C. Castaing: Topologie de la convergence uniforme sur les parties uniformément intégrables de L1 (E) et théorèmes de compacité faible dans certains espaces du type Köthe Orlicz, Semin. d’Anal. Convexe, Montpellier 10, Exposé 5 (1980).Google Scholar
  6. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  7. F. H. Clarke: Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Avanced Texts, Wiley, New York (1983).Google Scholar
  8. J. Couvreux: From the Mosco and slice convergences for convex integral functionals on Lp spaces, SIAM J. Control Optim. 39(1) (2000) 179–195.https://doi.org/10.1137/S105262349732222XGoogle Scholar
  9. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  10. J. Diestel, J. J. Uhl: Vector Measures, Math. Surveys Monographs 15, Amer. Math. Soc., Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  11. N. Dunford, J. T. Schwartz: Linear Operators. I. General Theory, John Wiley, New York (1958).Google Scholar
  12. I. Fonseca, G. Leoni: Modern Methods in the Calculus of Variations: Lp Spaces, Springer, New York (2007).Google Scholar
  13. E. Giner: Etudes sur les Fonctionnelles Intégrales, Thèse, Université de Pau (1985).Google Scholar
  14. E. Giner: Calmness properties and contingent subdifferential of an integral functional on Lebesgue spaces Lp, 1 ≤ p < ∞, Set-Valued Var. Anal. 17(3) (2009) 223–244.https://doi.org/10.1007/s11228-009-0116-1Google Scholar
  15. E. Giner: Fatou’s Lemma and lower epi-limits of integral functionals, J. Math. Anal. Appl. 394 (2012) 13–29.https://doi.org/10.1016/j.jmaa.2012.04.060Google Scholar
  16. E. Giner: A useful underestimate for the convergence of integral functionals, submitted.Google Scholar
  17. A. Grothendieck: Espaces Vectoriels Topologiques, Instituto de Matematica da Universidade de São Paulo, São Paulo (1954).Google Scholar
  18. F. Hiai, H. Umegaki: Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149–182.https://doi.org/10.1016/0047-259X(77)90037-9Google Scholar
  19. J. L. Joly: Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl., IX. Sér. 52 (1973) 421–441.Google Scholar
  20. J. L. Joly, F. De Thélin: Convergence of convex integrals in Lp spaces, J. Math. Anal. Appl. 54(1) (1976) 230–244.https://doi.org/10.1016/0022-247X(76)90247-XGoogle Scholar
  21. C. M. Marle: Mesures et Probabilités, Hermann, Paris (1974).Google Scholar
  22. U. Mosco: Convergence of convex sets and solutions of variational inequalities, Adv. Math. 3 (1969) 510–585.https://doi.org/10.1016/0001-8708(69)90009-7Google Scholar
  23. U. Mosco: On the continuity of the Young Fenchel transform, J. Math. Anal. Appl. 35 (1971) 518–535.https://doi.org/10.1016/0022-247X(71)90200-9Google Scholar
  24. J. Neveu: Bases Mathématiques du Calcul des Probabilités, Masson, Paris (1964).Google Scholar
  25. J. P. Penot: Book in preparation.Google Scholar
  26. J. P. Penot, C. Zălinescu: Continuity of usual operations and variational convergences, Set-Valued Anal. 11 (2003) 225–256.https://doi.org/10.1023/A:1024432532388Google Scholar
  27. J. P. Penot, C. Zălinescu: Continuity of the Legendre-Fenchel transform for some variational convergences, Optimization 53 (2004) 549–562.https://doi.org/10.1080/02331930412331329533Google Scholar
  28. R. T. Rockafellar: Convex integral functionals and duality, in: Contributions to Nonlinear Functional Analysis (Madison, 1971), E. H. Zarantonello (ed.), Academic Press, New York (1971) 215–236.https://doi.org/10.1016/B978-0-12-775850-3.50012-1Google Scholar
  29. R. T. Rockafellar: Integral functionals, normal integrands and measurable selections, in: Nonlinear Operators and the Calculus of Variations (Bruxelles, 1975), J. P. Gossez et al. (ed.), Lect. Notes in Math. 543, Springer (1976) 157–206.https://doi.org/10.1007/BFb0079944Google Scholar
  30. R. T. Rockafellar, J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  31. A. Salvadori: On the M-convergence for integral functionals on Lp (X), Atti Semin. Mat. Fis. Univ. Modena 33 (1984) 137–154.Google Scholar
  32. A. Salvadori: On the M-convergence for integral functionals on Lp (X), Semin. d’Anal. Convexe, Montpellier 15, Exposé 5 (1985), 25 p.Google Scholar
  33. A. Truffert: Propriété de Fatou Vitali: Applications à l’épiconvergence des fonctionnelles intégrales, D’aprés E. Giner, Semin. d’Anal. Convexe, Montpellier 13, Exposé 9 (1983).Google Scholar
  34. A. Truffert: Applications des méthodes de représentation intégrale et d’approximations inf-convolutives à l’épi convergence, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14(2) (1987) 169–197.Google Scholar
  35. M. Valadier: A natural supplement of L1 in the dual of L∞, Semin. d’Anal. Convexe, Montpellier 4, Exposé 13 (1974), 27 p.Google Scholar
  36. R. Wijsman: Convergence of sequences of convex sets, cones and functions II, Trans. Amer. Math. Soc. 123 (1966) 32–45.https://doi.org/10.1090/S0002-9947-1966-0196599-8Google Scholar

Suggested citation

E. Giner. “An Upper Bound for the Convergence of Integral Functionals.” Journal of Convex Analysis 20 (2013), No. 2, 355–376. https://doi.org/10.68381/jca20023

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