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Relaxation and regularity in the calculus of variations. (English) Zbl 1198.49034
Summary: 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.

##### 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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##### References:
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