Regularity of minimizers for second order variational problems in one independent variable doi:10.3934/dcds.2011.29.547
Christos Gavriel - Department of Electrical and Electronic Engineering, Imperial College London, SW7 2BT, United Kingdom (email) 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.
Keywords: Calculus of Variations, Minimizer Regularity, Autonomous Problems.
Received: September 2009; Revised: March 2010; Published: October 2010. |
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