Relaxation and regularity in the calculus of variations

In this work we prove that, if L(t,u,ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ, then any absolutely continuous solution to the variational problem is quasi-regular in the sense of Tonelli, i.e. 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 ξ, but not necessarily in t, or L is invariant under a group of C1 transformations (as in the Noetherʹs theorem). Without one of those further assumptions the solution could be not regular as shown by a recent example in Gratwick and Preiss (2010) [13]; our result is then optimal in this sense. Moreover, we improve the standard hypothesis used so far in Buttazzo et al. (1998) [1], Clarke and Vinter (1985) [5] and [6], Csörnyei et al. (2008) [7], Tonelli (1915) [15] which have been the Lipschitz continuity of L in u, locally uniform in ξ and t, and some growth condition in ξ. 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 Mariconda and Treu (2004) [14] to the non-autonomous case.