The 1-harmonic flow with values in a hyperoctant of the $N$-sphere

Abstract

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, ${\mathbb{S}}_{+}^{N-1}$, 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 ${\mathbb{S}}_{+}^{N-1}$ with respect to a metric which penalizes the closeness to their geodesic midpoint.