Existence of 1-harmonic map flow

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking value in the manifold. So far, except some global existence results for special choices of the target manifold, local existence of the flow was only known assuming the domain is a flat torus and the datum is small. 

 

Assuming the domain is convex, we prove local existence of the flow in Lipschitz class for large data and arbitrary target manifold. If the manifold has non-positive sectional curvature or the datum is small, the flow is shown to exist globally and become constant in finite time.

 

We also consider the case where the domain is a compact Riemannian manifold, solving the homotopy problem for 1-harmonic maps under some assumptions.  

 

This is joint work with L. Giacomelli and S. Moll