  We consider the group of smooth increasing diffeomorphisms Diff on the unit
interval endowed with the right-invariant $H^1$ metric. We compute the metric
completion of this space which appears to be the space of increasing maps of
the unit interval with boundary conditions at $0$ and $1$. We compute the
lower-semicontinuous envelope associated with the length minimizing geodesic
variational problem. We discuss the Eulerian and Lagrangian formulation of this
relaxation and we show that smooth solutions of the EPDiff equation are length
minimizing for short times.
