The 1-harmonic Flow with Values into a Hyper-octant of the N-sphere

We prove the existence of solutions to the 1-harmonic flow –i.e., the formal gradient flow of the total variation of a vector field with respect to the L-distance– from a domain of R into a hyper-octant of the N -dimensional unit sphere, S + , 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 semi-continuity argument which leads to a nontrivial and non-convex minimization problem: to find a shortest path between two points on S + with respect to a metric which penalizes the closeness to their geodesic midpoint.