On the velocities of Lagrangian minimizers

We consider minimizers for the natural time-dependent Lagrangian system in $\mathbb R^d$ with Lagrangian $L(x,v,t)=|v|^{\beta}/\beta-U(x,t)$, $\beta>1$, where $\beta>1$. For minimizers on a $T$ with one end-point fixed, we prove that the absolute values of velocities are bounded by $K\log^{2/\beta}T$, provided that the potential $U(x,t)$ and its gradient are uniformly bounded. We also show that the above estimate is asymptotically sharp.