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

Abstract : 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.
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Contributor : François-Xavier Vialard <>
Submitted on : Friday, June 21, 2019 - 9:56:12 AM
Last modification on : Thursday, February 13, 2020 - 2:02:18 PM


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  • HAL Id : hal-02161686, version 1
  • ARXIV : 1906.09139


Simone Di Marino, Andrea Natale, Rabah Tahraoui, François-Xavier Vialard. Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric. 2019. ⟨hal-02161686⟩



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