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. For the resulting PDE, I
will introduce a notion of solution and I will discuss existence and
uniqueness results for two specific manifolds: the hyper-octant of an
N-dimensional sphere and a connected sub-arc of a regular Jordan curve. I
will also present possible extensions to general manifolds, together with
related open questions and conjectures.
Based on joint works with Agnese Di Castro, José Mazòn, and Salvador Moll.

http://cvgmt.sns.it/seminar/546/