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A minimal approach to the theory of global attractors
A direct proof of the Tonelli's partial regularity result
1. | Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco,28049 Madrid |
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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