Differential and Integral Equations

On the Tonelli's Partial Regularity

Alessandro Ferriero

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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., $L(t,u,\xi)=L(u,\xi)$.

Article information

Differential Integral Equations, Volume 26, Number 1/2 (2013), 1-9.

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49B05 49A05 449C05 35J20: Variational methods for second-order elliptic equations


Ferriero, Alessandro. On the Tonelli's Partial Regularity. Differential Integral Equations 26 (2013), no. 1/2, 1--9. https://projecteuclid.org/euclid.die/1355867504

Export citation