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April  2011, 29(2): 547-557. doi: 10.3934/dcds.2011.29.547

## Regularity of minimizers for second order variational problems in one independent variable

 1 Department of Electrical and Electronic Engineering, Imperial College London, SW7 2BT, United Kingdom, United Kingdom

Received  September 2009 Revised  March 2010 Published  October 2010

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.
Citation: Christos Gavriel, Richard Vinter. Regularity of minimizers for second order variational problems in one independent variable. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 547-557. doi: 10.3934/dcds.2011.29.547
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