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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A regularity theory for variational problems with higher order derivatives
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by F. H. Clarke and R. B. Vinter
Trans. Amer. Math. Soc. 320 (1990), 227-251
DOI: https://doi.org/10.1090/S0002-9947-1990-0970266-6

Abstract:

We consider problems in the calculus of variations in one independent variable and where the Lagrangian involves derivatives up to order $N$, $N \ge 1$. Existence theory supplies mild hypotheses under which there are minimizers for such problems, but they need to be strengthened for standard necessary conditions to apply. For problems with $N > 1$, this paper initiates investigation of regularity properties, and associated necessary conditions, which obtain strictly under the hypotheses of existence theory. It is shown that the $N$th derivative of a minimizer is locally essentially bounded off a closed set of zero measure, the set of "points of bad behaviour". Additional hypotheses are shown to exclude occurrence of points of bad behaviour. Finally a counter example suggests respects in which problems with $N > 1$ exhibit pathologies not present in the $N = 1$ case.
References
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Bibliographic Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 320 (1990), 227-251
  • MSC: Primary 49A21; Secondary 49B21
  • DOI: https://doi.org/10.1090/S0002-9947-1990-0970266-6
  • MathSciNet review: 970266