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Abstract
We consider autonomous, second order problems in the calculus of variations in one independent variable. For analogous first order problems it is known that, under standard hypotheses of existence theory and a local boundedness condition on the Lagrangian, minimizers over $W^{1,1}$ have bounded first derivatives ($W^{1,\infty}$ regularity prevails). For second order problems one might expect, by analogy, that minimizers would have bounded second derivatives ($W^{2,\infty}$ regularity) under the standard existence hypotheses $(HE)$ for second order problems, supplemented by a local boundedness condition. A counter-example, however, indicates that this is not the case. In earlier work, $W^{2, \infty}$ regularity has been established for these problems under $(HE)$ and additional 'integrability' hypotheses on derivatives of the Lagrangian, evaluated along the minimizer. We show that these additional hypotheses can be significantly reduced. The proof techniques employed depend on a combination of the application of a change of independent variable and of extensions to Tonelli regularity theory proved by Clarke and Vinter.
Mathematics Subject Classification: Primary: 49J10, 49J30; Secondary: 49J99.
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