Open Access
January/February 2013 On the Tonelli's Partial Regularity
Alessandro Ferriero
Differential Integral Equations 26(1/2): 1-9 (January/February 2013). DOI: 10.57262/die/1355867504


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


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Alessandro Ferriero. "On the Tonelli's Partial Regularity." Differential Integral Equations 26 (1/2) 1 - 9, January/February 2013.


Published: January/February 2013
First available in Project Euclid: 18 December 2012

zbMATH: 1289.49041
MathSciNet: MR3058695
Digital Object Identifier: 10.57262/die/1355867504

Primary: 35J20 , 449C05 , 49A05 , 49B05

Rights: Copyright © 2013 Khayyam Publishing, Inc.

Vol.26 • No. 1/2 • January/February 2013
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