The 1-harmonic flow

The $1$-harmonic flow is the formal gradient flow --with respect to the $L^2$-distance-- of the total variation of a manifold-valued unknown function. The problem originates from image processing and has an intrinsic analytical interest as prototype of constrained and vector-valued evolution equations in $BV$-spaces. I will introduce a notion of solution and I will present existence (and, in some cases, uniqueness) results when the target manifold is either the hyper-octant of a sphere or a connected subarc of a regular Jordan curve. I will also discuss a work in progress concerning local/global-in-time well-posedness in Lipschitz spaces. As all of the above is just a first, tentative step into a rather uncharted territory, I will conclude by highlighting a few basic open questions.