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Transactions of the American Mathematical Society

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

 

 

A regularity theory for variational problems with higher order derivatives


Authors: F. H. Clarke and R. B. Vinter
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
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0970266-6
Article copyright: © Copyright 1990 American Mathematical Society