*Published Paper*

**Inserted:** 9 jul 2012

**Last Updated:** 4 feb 2021

**Journal:** SIAM Journal Math. Anal.

**Volume:** 45

**Pages:** 1723-1740

**Year:** 2013

**Links:**
DOI

**Abstract:**

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 $\mathbb R^m$ into a geodesically convex subset of an $N$-sphere. For such 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**, 1471-1498 and in X. Feng, Calc. Var. Partial Differential Equations **37**, 111-139 are also discussed.

**Keywords:**
harmonic maps, Total variation flow, Nonlinear parabolic equations.