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On the Brezis–Lieb lemma without pointwise convergence

  • Adimurthi
  • Cyril TintarevEmail author
Article

Abstract

Brezis–Lieb lemma is an improvement of Fatou Lemma that evaluates the gap between the integral of a functional sequence and the integral of its pointwise limit. The paper proves some analogs of Brezis–Lieb lemma without assumption of convergence almost everywhere. While weak convergence alone brings no conclusive estimates, a lower bound for the gap is found in L p , p ≥ 3, under condition of weak convergence and weak convergence in terms of the duality mapping. We prove that the restriction on p is necessary and prove few related inequalities in connection to weak convergence.

Mathematics Subject Classification

Primary 49J45 49J49  Secondary 35B27 46B99 

References

  1. 1.
    Brezis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Moreira D.R., Teixeira E.V.: Weak convergence under nonlinearities. An. Acad. Bras. Cienc. 75(1), 9–19 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Moreira, D.R., Teixeira, E.V.: On the behavior of weak convergence under nonlinearities and applications. Proc. Am. Math. Soc. 133, 1647–1656 (2005) (electronic)Google Scholar
  4. 4.
    Opial Z.: Weak Convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Solimini, S., Tintarev, C.: Concentration analysis in Banach spaces. Commun. Contemp. Math. (to appear)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.TIFR CAMBangaloreIndia
  2. 2.Uppsala UniversityUppsalaSweden

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