On the Tonelli's Partial Regularity

Abstract

In this work we prove that the symmetries of a Lagrangian function $L$ play an important role in the regularity of the solutions to its associated variational problem. More precisely, we prove that any absolutely continuous solution to $$\min \Big \{\int_a^b L(t,u(t),\dot u(t))\,dt: u\in{\bf W}_0^{1,1}(a,b)\Big \}$$ is regular in the sense of Tonelli, that is, has extended-values continuous derivative, if $L(t,u,\xi)$ is strictly convex in $\xi$, continuous in $t$ and $u$, and is invariant under a group of ${\bf C}^1$ transformations as in Noether's theorem. Our proof does not require $L$ to be Lipschitz continuous with respect to $u$, which is the standard hypothesis [2, 4, 5, 6, 8, 13] for proving this kind of regularity. As a corollary of our main result, we then obtain the Tonelli's partial regularity result without assuming any regularity of $L$ in $u$ (more than continuity) in the autonomous case, i.e., $L(t,u,\xi)=L(u,\xi)$.