Vol. 7, No. 3, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 2, 253–512
Issue 1, 1–252

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
Cover
About the Cover
Editorial Board
Editors’ Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
The 1-harmonic flow with values in a hyperoctant of the $N$-sphere

Lorenzo Giacomelli, Jose M. Mazón and Salvador Moll

Vol. 7 (2014), No. 3, 627–671
Abstract

We prove the existence of solutions to the 1-harmonic flow — that is, the formal gradient flow of the total variation of a vector field with respect to the L2-distance — from a domain of m into a hyperoctant of the N-dimensional unit sphere, S+N1, under homogeneous Neumann boundary conditions. In particular, we characterize the lower-order term appearing in the Euler–Lagrange formulation in terms of the “geodesic representative” of a BV-director field on its jump set. Such characterization relies on a lower semicontinuity argument which leads to a nontrivial and nonconvex minimization problem: to find a shortest path between two points on S+N1 with respect to a metric which penalizes the closeness to their geodesic midpoint.

Keywords
harmonic flows, total variation flow, nonlinear parabolic systems, lower semicontinuity and relaxation, nonconvex variational problems, geodesics, Riemannian manifolds with boundary, image processing
Mathematical Subject Classification 2010
Primary: 35K55, 49Q20, 53C22, 35K67, 35K92
Secondary: 49J45, 53C44, 58E20, 68U10
Milestones
Received: 18 April 2013
Accepted: 27 November 2013
Published: 18 June 2014
Authors
Lorenzo Giacomelli
SBAI Department
Sapienza University of Rome
Via Scarpa, 16
I-00161 Roma
Italy
Jose M. Mazón
Departament d’Anàlisi Matemàtica
Universitat de València
Dr. Moliner, 50
46100 Burjassot
Spain
Salvador Moll
Departament d’Anàlisi Matemàtica
Universitat de València
Dr. Moliner, 50
46100 Burjassot
Spain