The 1-harmonic flow with values into a smooth planar curve

We introduce a notion of solution to the 1-harmonic flow –i.e., the formal gradient flow of the total variation with respect to the L^2-distance– from a domain of R^m into a connected subset of the image of a smooth Jordan curve. For such notion, we establish existence and uniqueness of solutions to the homogeneous Neumann problem. We also discuss a consistent notion of solution when the target space is a smooth (n − 1)-dimensional manifold whose geodesics are unique, presenting conjectures and open questions related to it.