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.
Keywords
harmonic flows, total variation flow, nonlinear parabolic
systems, lower semicontinuity and relaxation, nonconvex
variational problems, geodesics, Riemannian manifolds with
boundary, image processing