Regularity of minimizers for higher order variational problems in one independent variable

Abstract

This paper concerns problems in the calculus of variations in one independent variable, when the Lagrangian depends on derivates of the state trajectories up to order N. For first order problems (N = 1) it is well known that, under standard hypotheses of existence theory and a local boundedness condition on the Lagrangian, minimizers have uniformly bounded first derivatives. These properties are of interest, because they ensure validity of necessary conditions for analysing minimizers, such as the Euler Lagrange equation, and give insights in appropriate descritization schemes for numerical solution. For Nth order problems one might expect, by analogy with the N = 1 case, that minimizers would have uniformly bounded Nth order derivatives. This is not the case in general, however, as illustrated by known counter examples. To guarantee boundedness of the Nth order derivatives it has been found necessary to introduce additional integrability hypotheses on derivatives of the Lagrangian, evaluated along the minimizer. We show that the additional hypotheses, previously imposed to guarantee uniform boundedness of the highest order derivatives, can be signicantly reduced. This paper improves in particular on recent work on the boundedness of the second order derivates for second order problems, based on an analysis specific to the N = 2 case.