## 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
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## 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