A direct proof of the Tonelli's partial regularity result

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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

Abstract
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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].

Keywords: quasi-linear elliptic differential equations., regularity, Variational problems.

Mathematics Subject Classification: Primary: 49B05, 49A05, 49C05, 35J2.

Citation: Alessandro Ferriero. A direct proof of the Tonelli's partial regularity result. Discrete & Continuous Dynamical Systems - A, 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. 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 (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. 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. 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. 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. 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. 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. doi: 10.1137/S0363012903437721. Google Scholar
[9]	A. Ferriero, Relaxation and regularity in the calculus of variations,, J. Differential Equations, 249 (2010), 2548. 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. 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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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. 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 (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. 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. 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. 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. 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. 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. doi: 10.1137/S0363012903437721. Google Scholar
[9]	A. Ferriero, Relaxation and regularity in the calculus of variations,, J. Differential Equations, 249 (2010), 2548. 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. 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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