# American Institute of Mathematical Sciences

June  2012, 32(6): 2089-2099. doi: 10.3934/dcds.2012.32.2089

## A direct proof of the Tonelli's partial regularity result

Received  April 2011 Revised  July 2011 Published  February 2012

The aim of this work is to give a simple proof of the Tonelli's partial regularity result which states that any absolutely continuous solution to the variational problem $$\min\left\{\int_a^b L(t,u(t),\dot u(t))dt: u\in{\bf W}_0^{1,1}(a,b)\right\}$$ has extended-values continuous derivative if the Lagrangian function $L(t,u,\xi)$ is strictly convex in $\xi$ and Lipschitz continuous in $u$, locally uniformly in $\xi$ (but not in $t$). Our assumption is weaker than the one used in [2, 4, 5, 6, 13] since we do not require the Lipschitz continuity of $L$ in $u$ to be locally uniform in $t$, and it is optimal as shown by the example in [12].
Citation: Alessandro Ferriero. A direct proof of the Tonelli's partial regularity result. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2089-2099. doi: 10.3934/dcds.2012.32.2089
##### References:
 [1] J. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mech. Anal., 90 (1985), 325-388. doi: 10.1007/BF00276295. [2] G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-Dimensional Variational Problems. An Introduction," Oxford Lecture Series in Mathematics and its Applications, 15, The Clarendon Press, Oxford University Press, New York, 1998. [3] A. Cellina, A. Ferriero and E. M. Marchini, Reparameterizations and approximate values of integrals of the calculus of variations, J. Diff. Equations, 193 (2003), 374-384. doi: 10.1016/S0022-0396(02)00176-6. [4] F. H. Clarke and R. B. Vinter, Existence and regularity in the small in the calculus of variations, J. Diff. Equations, 59 (1985), 336-354. doi: 10.1016/0022-0396(85)90145-7. [5] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Am. Math. Soc., 289 (1985), 73-98. doi: 10.1090/S0002-9947-1985-0779053-3. [6] M. Csörnyei, B. Kirchheim, T. O'Neil, D. Preiss and S. Winter, Universal singular sets in the calculus of variations, Arch. Rat. Mech. Anal., 190 (2008), 371-424. doi: 10.1007/s00205-008-0142-4. [7] A. M. Davie, Singular minimizers in the calculus of variations in one dimension, Arch. Rat. Mech. Anal., 101 (1988), 161-177. doi: 10.1007/BF00251459. [8] A. Ferriero, The approximation of higher-order integrals of the calculus of variations and the Lavrentiev phenomenon, SIAM J. Control Optim., 44 (2005), 99-110. doi: 10.1137/S0363012903437721. [9] A. Ferriero, Relaxation and regularity in the calculus of variations, J. Differential Equations, 249 (2010), 2548-2560. doi: 10.1016/j.jde.2010.06.013. [10] A. Ferriero, On the Tonelli's partial regularity, preprint, 2008. [11] A. Ferriero and E. M. Marchini, On the validity of the Euler-Lagrange equation, J. Math. Anal. Appl., 304 (2005), 356-369. doi: 10.1016/j.jmaa.2004.09.029. [12] R. Gratwick and D. Preiss, A one-dimensional variational problem with continuous Lagrangian and singular minimizer, preprint, 2010. [13] L. Tonelli, Sur un méthode directe du calcul des variations, Rend. Circ. Mat. Palermo., 39 (1915).

show all references

##### References:
 [1] J. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rat. Mech. Anal., 90 (1985), 325-388. doi: 10.1007/BF00276295. [2] G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-Dimensional Variational Problems. An Introduction," Oxford Lecture Series in Mathematics and its Applications, 15, The Clarendon Press, Oxford University Press, New York, 1998. [3] A. Cellina, A. Ferriero and E. M. Marchini, Reparameterizations and approximate values of integrals of the calculus of variations, J. Diff. Equations, 193 (2003), 374-384. doi: 10.1016/S0022-0396(02)00176-6. [4] F. H. Clarke and R. B. Vinter, Existence and regularity in the small in the calculus of variations, J. Diff. Equations, 59 (1985), 336-354. doi: 10.1016/0022-0396(85)90145-7. [5] F. H. Clarke and R. B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Am. Math. Soc., 289 (1985), 73-98. doi: 10.1090/S0002-9947-1985-0779053-3. [6] M. Csörnyei, B. Kirchheim, T. O'Neil, D. Preiss and S. Winter, Universal singular sets in the calculus of variations, Arch. Rat. Mech. Anal., 190 (2008), 371-424. doi: 10.1007/s00205-008-0142-4. [7] A. M. Davie, Singular minimizers in the calculus of variations in one dimension, Arch. Rat. Mech. Anal., 101 (1988), 161-177. doi: 10.1007/BF00251459. [8] A. Ferriero, The approximation of higher-order integrals of the calculus of variations and the Lavrentiev phenomenon, SIAM J. Control Optim., 44 (2005), 99-110. doi: 10.1137/S0363012903437721. [9] A. Ferriero, Relaxation and regularity in the calculus of variations, J. Differential Equations, 249 (2010), 2548-2560. doi: 10.1016/j.jde.2010.06.013. [10] A. Ferriero, On the Tonelli's partial regularity, preprint, 2008. [11] A. Ferriero and E. M. Marchini, On the validity of the Euler-Lagrange equation, J. Math. Anal. Appl., 304 (2005), 356-369. doi: 10.1016/j.jmaa.2004.09.029. [12] R. Gratwick and D. Preiss, A one-dimensional variational problem with continuous Lagrangian and singular minimizer, preprint, 2010. [13] L. Tonelli, Sur un méthode directe du calcul des variations, Rend. Circ. Mat. Palermo., 39 (1915).
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