THE 1-HARMONIC FLOW WITH VALUES IN A HYPEROCTANT OF THE N-SPHERE

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, 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 S-+(N-1) with respect to a metric which penalizes the closeness to their geodesic midpoint.