The 1-harmonic flow with values into S-1

We introduce a notion of solution for the 1-harmonic flow, i.e., the formal gradient flow of the total variation functional with respect to the L-2-distance, from a domain of R-m into a geodesically convex subset of an N-sphere. For such a notion, under homogeneous Neumann boundary conditions, we prove both existence and uniqueness of solutions when the target space is a semicircle and the existence of solutions when the target space is a circle and the initial datum has no jumps of an "angle" larger than pi. Earlier results in [J. W. Barrett, X. Feng, and A. Prohl, SIAM J. Math. Anal., 40 (2008), pp. 1471-1498] and [X. Feng, Calc. Var. Partial Differential Equations, 37 (2011), pp. 111-139] are also discussed.