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2014 The 1-harmonic flow with values in a hyperoctant of the $N$-sphere
Lorenzo Giacomelli, Jose Mazón, Salvador Moll
Anal. PDE 7(3): 627-671 (2014). DOI: 10.2140/apde.2014.7.627


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.


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Lorenzo Giacomelli. Jose Mazón. Salvador Moll. "The 1-harmonic flow with values in a hyperoctant of the $N$-sphere." Anal. PDE 7 (3) 627 - 671, 2014.


Received: 18 April 2013; Accepted: 27 November 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1295.35260
MathSciNet: MR3227428
Digital Object Identifier: 10.2140/apde.2014.7.627

Primary: 35K55, 35K67, 35K92, 49Q20, 53C22
Secondary: 49J45, 53C44, 58E20, 68U10

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 3 • 2014
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