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A direct proof of the Tonelli's partial regularity result

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  • The aim of this work is to give a simple proof of the Tonelli's partial regularity result which states that any absolutely continuous solution to the variational problem $$\min\left\{\int_a^b L(t,u(t),\dot u(t))dt: u\in{\bf W}_0^{1,1}(a,b)\right\}$$ has extended-values continuous derivative if the Lagrangian function $L(t,u,\xi)$ is strictly convex in $\xi$ and Lipschitz continuous in $u$, locally uniformly in $\xi$ (but not in $t$). Our assumption is weaker than the one used in [2, 4, 5, 6, 13] since we do not require the Lipschitz continuity of $L$ in $u$ to be locally uniform in $t$, and it is optimal as shown by the example in [12].
    Mathematics Subject Classification: Primary: 49B05, 49A05, 49C05, 35J20.


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