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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 |
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Vol. 7 (2014), No. 3, 627–671
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Abstract |
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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 -distance — from a domain of into a hyperoctant of the -dimensional unit sphere, , 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 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
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Mathematical Subject Classification 2010
Primary: 35K55, 49Q20, 53C22, 35K67, 35K92
Secondary: 49J45, 53C44, 58E20, 68U10
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Milestones
Received: 18 April 2013
Accepted: 27 November 2013
Published: 18 June 2014
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Authors |
| Lorenzo Giacomelli | |
| SBAI Department Sapienza University of Rome Via Scarpa, 16 I-00161 Roma Italy |
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| Jose M. Mazón | |
| Departament d’Anàlisi
Matemàtica Universitat de València Dr. Moliner, 50 46100 Burjassot Spain |
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| Salvador Moll | |
| Departament d’Anàlisi
Matemàtica Universitat de València Dr. Moliner, 50 46100 Burjassot Spain |
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