A regularity theory for variational problems with higher order derivatives

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.