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Direct Methods in Variational Field Theory

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Abstract

We show that the Weierstrass–Hilbert classical field theory can be strengthened. Namely, for each extremal field, it is true that if an extremal is an element of the field then a minimum is attained in the class of Sobolev functions with the same boundary data as for the extremal and with graphs in the set covered by the field. This result remains valid if one of the extremals is singular. If there is a field containing more than one singular extremal then each of these extremals defines the minimization problem having no solution in the class of Lipschitz functions with graphs in the set covered by the field.

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Funding

The study was carried out in the framework of the State Task (Project FWNF–2022–0008) and supported by the RFBR (Project 18–01–00649).

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Authors and Affiliations

  1. University of Edinburgh, Edinburgh, United Kingdom

    R. Gratwick

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    M. A. Sychev

Authors
  1. R. Gratwick
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  2. M. A. Sychev
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Corresponding author

Correspondence to R. Gratwick.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 5, pp. 1027–1034. https://doi.org/10.33048/smzh.2022.63.505

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Gratwick, R., Sychev, M.A. Direct Methods in Variational Field Theory. Sib Math J 63, 862–867 (2022). https://doi.org/10.1134/S0037446622050056

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  • DOI: https://doi.org/10.1134/S0037446622050056

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