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

  • Chapter
Recent Developments in Well-Posed Variational Problems

Part of the book series: Mathematics and Its Applications ((MAIA,volume 331))

Abstract

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;

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Via Buonarroti, 2, 56127, Pisa, Italy

    G. Buttazzo & M. Belloni

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  1. G. Buttazzo
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  2. M. Belloni
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Editors and Affiliations

  1. Department of Mathematics, University of Milan, Milan, Italy

    Roberto Lucchetti

  2. Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Julian Revalski

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Buttazzo, G., Belloni, M. (1995). A Survey on Old and Recent Results about the Gap Phenomenon in the Calculus of Variations. In: Lucchetti, R., Revalski, J. (eds) Recent Developments in Well-Posed Variational Problems. Mathematics and Its Applications, vol 331. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8472-2_1

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  • DOI: https://doi.org/10.1007/978-94-015-8472-2_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4578-2

  • Online ISBN: 978-94-015-8472-2

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