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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 and Continuous Dynamical Systems, 2011, 29 (2) : 547-557. doi: 10.3934/dcds.2011.29.547
##### References:
 [1] A. V. Sarychev and D. F. Torres, Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics, Appl. Math. Optim., 41 (2000), 237-254. doi: doi:10.1007/s002459911013. [2] D. R. Smith, "Variational Methods in Optimization," Prentice Hall, Englewood Cliffs, N.J., 1974. [3] F. H. Clarke, L. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, vol. 178, Springer Verlag, New York, 1998. [4] F. H. Clarke, "Methods of Dynamic and Nonsmooth Optimization," Regional Conference Series in Applied Mathematics, SIAM Publications, PA, 1989. [5] F. H. Clarke, Necessary conditions in dynamic optimization, Memoirs of the Amer. Math. Soc., 816 (2005), 1-110. [6] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), 73-98. [7] F. H. Clarke and R. B. Vinter, A regularity theory for variational problems with higher order derivatives, Trans. Amer. Math. Soc., 320 (1990), 227-251. doi: doi:10.2307/2001759. [8] F. H. Clarke and R. B. Vinter, On the conditions under which the Euler equation or the Maximum principle hold, Appl. Math. Optim, 12 (1983), 73-79. doi: doi:10.1007/BF01449034. [9] L. Ambrosio, O. Ascenti and G. Butazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. An. Applic., 142 (1989), 301-316. doi: doi:10.1016/0022-247X(89)90001-2. [10] L. C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory," Saunders, Philadelphia, 1969. [11] L. Tonelli, "Fondamenti di Calcolo delle Variazioni," vol. 1 and 2, Zanichelli, Bologna, 1921, 1923. [12] R. B. Vinter, "Optimal Control," Birkhauser, Boston, 2000. [13] R. T. Rockafellar and R. J.-W. Wets, "Variational Analysis, Grundlehren der Mathematischen Wissenschaften," vol. 317, Springer Verlag, New York, 1998. [14] R. T. Rockafellar, Existence theorems for general control problems of Bolza and Lagrange, Advances in Math., 15 (1975), 312-333. doi: doi:10.1016/0001-8708(75)90140-1.

show all references

##### References:
 [1] A. V. Sarychev and D. F. Torres, Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics, Appl. Math. Optim., 41 (2000), 237-254. doi: doi:10.1007/s002459911013. [2] D. R. Smith, "Variational Methods in Optimization," Prentice Hall, Englewood Cliffs, N.J., 1974. [3] F. H. Clarke, L. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, vol. 178, Springer Verlag, New York, 1998. [4] F. H. Clarke, "Methods of Dynamic and Nonsmooth Optimization," Regional Conference Series in Applied Mathematics, SIAM Publications, PA, 1989. [5] F. H. Clarke, Necessary conditions in dynamic optimization, Memoirs of the Amer. Math. Soc., 816 (2005), 1-110. [6] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc., 289 (1985), 73-98. [7] F. H. Clarke and R. B. Vinter, A regularity theory for variational problems with higher order derivatives, Trans. Amer. Math. Soc., 320 (1990), 227-251. doi: doi:10.2307/2001759. [8] F. H. Clarke and R. B. Vinter, On the conditions under which the Euler equation or the Maximum principle hold, Appl. Math. Optim, 12 (1983), 73-79. doi: doi:10.1007/BF01449034. [9] L. Ambrosio, O. Ascenti and G. Butazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. An. Applic., 142 (1989), 301-316. doi: doi:10.1016/0022-247X(89)90001-2. [10] L. C. Young, "Lectures on the Calculus of Variations and Optimal Control Theory," Saunders, Philadelphia, 1969. [11] L. Tonelli, "Fondamenti di Calcolo delle Variazioni," vol. 1 and 2, Zanichelli, Bologna, 1921, 1923. [12] R. B. Vinter, "Optimal Control," Birkhauser, Boston, 2000. [13] R. T. Rockafellar and R. J.-W. Wets, "Variational Analysis, Grundlehren der Mathematischen Wissenschaften," vol. 317, Springer Verlag, New York, 1998. [14] R. T. Rockafellar, Existence theorems for general control problems of Bolza and Lagrange, Advances in Math., 15 (1975), 312-333. doi: doi:10.1016/0001-8708(75)90140-1.
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