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

A direct proof of the Tonelli's partial regularity result

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco,28049 Madrid

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Ferriero, On the Tonelli's partial regularity, preprint, 2008. Google Scholar

[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.  Google Scholar

[12]

R. Gratwick and D. Preiss, A one-dimensional variational problem with continuous Lagrangian and singular minimizer, preprint, 2010. Google Scholar

[13]

L. Tonelli, Sur un méthode directe du calcul des variations, Rend. Circ. Mat. Palermo., 39 (1915). Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Ferriero, On the Tonelli's partial regularity, preprint, 2008. Google Scholar

[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.  Google Scholar

[12]

R. Gratwick and D. Preiss, A one-dimensional variational problem with continuous Lagrangian and singular minimizer, preprint, 2010. Google Scholar

[13]

L. Tonelli, Sur un méthode directe du calcul des variations, Rend. Circ. Mat. Palermo., 39 (1915). Google Scholar

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