Relaxation and regularity in the calculus of variations.

*(English)*Zbl 1198.49034Summary: We prove that, if \(L(t,u,\xi )\) is a continuous function in \(t\) and \(u\), Borel measurable in \(\xi \), with bounded non-convex pieces in \(\xi \), then any absolutely continuous solution \(\bar u\) to the variational problem

\[ \min \left\{\int ^b_a L(t, u(t), \dot u(t))dt: u \in W^{1,1}_0 (a, b)\right\} \] is quasi-regular in the sense of Tonelli, i.e. \(\bar u\) is locally Lipschitz on an open set of full measure of \([a,b]\), under the further assumption that either \(L\) is Lipschitz continuous in \(u\), locally uniformly in \(\xi \), but not necessarily in \(t\), or \(L\) is invariant under a group of \(C^{1}\) transformations (as in Noether’s theorem). Without one of those further assumptions the solution could be not regular as shown by a recent example in Gratwick and Preiss (2010); our result is then optimal in this sense. Moreover, we improve the standard hypothesis used so far in G. Buttazzo, M. Giaquinta and S. Hildebrandt [One-dimensional variational problems. An introduction. Oxford: Clarendon Press (1998; Zbl 0915.49001)], F. H. Clarke and R. B. Vinter [J. Differ. Equations 59, No. 3, 336–354 (1985; Zbl 0727.49003) and Trans. Am. Math. Soc. 289, 73–98 (1985; Zbl 0563.49009)], M. Csörnyei, B. Kirchheim, T. C. O’Neil, D. Preiss and S. Winter, Arch. Ration. Mech. Anal. 190, No. 3, 371–424 (2008; Zbl 1218.49049)], Tonelli [Palermo Rend. 39, 233–264 (1915; JFM 45.0615.02)] which have been the Lipschitz continuity of \(L\) in \(u\), locally uniform in \(\xi \) and \(t\), and some growth condition in \(\xi \).

We also show that the relaxed and the original problem have the same solutions (without assuming any of the two further assumptions above). This extends a result in C. Mariconda and G. Treu [ESAIM, Control Optim. Calc. Var. 10, 201–210 (2004; Zbl 1072.49012)] to the non-autonomous case.

\[ \min \left\{\int ^b_a L(t, u(t), \dot u(t))dt: u \in W^{1,1}_0 (a, b)\right\} \] is quasi-regular in the sense of Tonelli, i.e. \(\bar u\) is locally Lipschitz on an open set of full measure of \([a,b]\), under the further assumption that either \(L\) is Lipschitz continuous in \(u\), locally uniformly in \(\xi \), but not necessarily in \(t\), or \(L\) is invariant under a group of \(C^{1}\) transformations (as in Noether’s theorem). Without one of those further assumptions the solution could be not regular as shown by a recent example in Gratwick and Preiss (2010); our result is then optimal in this sense. Moreover, we improve the standard hypothesis used so far in G. Buttazzo, M. Giaquinta and S. Hildebrandt [One-dimensional variational problems. An introduction. Oxford: Clarendon Press (1998; Zbl 0915.49001)], F. H. Clarke and R. B. Vinter [J. Differ. Equations 59, No. 3, 336–354 (1985; Zbl 0727.49003) and Trans. Am. Math. Soc. 289, 73–98 (1985; Zbl 0563.49009)], M. Csörnyei, B. Kirchheim, T. C. O’Neil, D. Preiss and S. Winter, Arch. Ration. Mech. Anal. 190, No. 3, 371–424 (2008; Zbl 1218.49049)], Tonelli [Palermo Rend. 39, 233–264 (1915; JFM 45.0615.02)] which have been the Lipschitz continuity of \(L\) in \(u\), locally uniform in \(\xi \) and \(t\), and some growth condition in \(\xi \).

We also show that the relaxed and the original problem have the same solutions (without assuming any of the two further assumptions above). This extends a result in C. Mariconda and G. Treu [ESAIM, Control Optim. Calc. Var. 10, 201–210 (2004; Zbl 1072.49012)] to the non-autonomous case.

##### MSC:

49N60 | Regularity of solutions in optimal control |

35J20 | Variational methods for second-order elliptic equations |

49M20 | Numerical methods of relaxation type |

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\textit{A. Ferriero}, J. Differ. Equations 249, No. 10, 2548--2560 (2010; Zbl 1198.49034)

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##### References:

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[13] | R. Gratwick, D. Preiss, A one-dimensional variational problem with continuous Lagrangian and singular minimizer, preprint, 2010. · Zbl 1266.70028 |

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