Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric
Optimization and Control
2019-06-24 v1 Analysis of PDEs
Abstract
We consider the group of smooth increasing diffeomorphisms Diff on the unit interval endowed with the right-invariant 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 and . 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.
Keywords
Cite
@article{arxiv.1906.09139,
title = {Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric},
author = {Simone Di Marino and Andrea Natale and Rabah Tahraoui and François-Xavier Vialard},
journal= {arXiv preprint arXiv:1906.09139},
year = {2019}
}