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

Di Marino, Simone; Natale, Andrea; Tahraoui, Rabah; Vialard, François-Xavier (2019-06), Metric completion of Diff([0,1]) with the H1 right-invariant metric. https://basepub.dauphine.fr/handle/123456789/19402

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Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-02161686
Date
2019-06
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Published in
Paris
Pages
18
Metadata
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Author(s)
Di Marino, Simone
INDAM Unit [Roma]
Natale, Andrea
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tahraoui, Rabah
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Vialard, François-Xavier
Laboratoire d'Informatique Gaspard-Monge [LIGM]
Abstract (EN)
We consider the group of smooth increasing diffeomorphisms Diff on the unit interval endowed with the right-invariant H1 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.
Subjects / Keywords
Diff; Metric completion