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

Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric