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.
"The 1-harmonic flow with values in a hyperoctant of the $N$-sphere." Anal. PDE 7 (3) 627 - 671, 2014. https://doi.org/10.2140/apde.2014.7.627