#### Vol. 7, No. 3, 2014

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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 ${L}^{2}$-distance — from a domain of ${ℝ}^{m}$ into a hyperoctant of the $N$-dimensional unit sphere, ${\mathbb{S}}_{+}^{N-1}$, 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 ${\mathbb{S}}_{+}^{N-1}$ 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