A Survey on Old and Recent Results about the Gap Phenomenon in the Calculus of Variations

  • G. Buttazzo
  • M. Belloni
Part of the Mathematics and Its Applications book series (MAIA, volume 331)


The term Lavrentiev phenomenon refers to the quite surprising feature of some functionals of the calculus of variations to possess different infima if considered on the full class of admissible functions and on the smaller class of regular admissible functions. The first example was found by Lavrentiev [33] in 1926, and since then many authors have considered this problem from different point of view (see References). In particular:
  1. (a)

    Manià [35], Heinricher and Mizel [29] simplified the original Lavrentiev example;

  2. (b)

    Ball and Mizel [6], [7], Davie [23], Loewen [34] demonstrated that the phenomenon can occur even with fully regular integrands;

  3. (c)

    Angell [4], Cesari [16], Clarke and Vinter [21] devised conditions which forestall occurrence of the phenomenon;

  4. (d)

    Ball and Mizel [7], Heinricher and Mizel [29] sharpened the specification of the precise dense subclass of admissible functions for which the Lavrentiev gap occurs;

  5. (e)

    Heinricher and Mizel [28], [30] presented an analogous gap phenomenon in stochastic control and in certain deterministic Bolza problems;

  6. (f)

    Ball and Mizel [7] investigated about the presence of the Lavrentiev phenomenon in certain problems of nonlinear elasticity, where it seems related to the formation of fractures;



Homogeneous Case Admissible Function Lipschitz Continuous Function Stochastic Control Problem Autonomous Functional 
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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • G. Buttazzo
    • 1
  • M. Belloni
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

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