Abstract

We will address a geometric evolution, corresponding to a curvature which is non-local and singular at small scales. It corresponds to the first variation of an energy proposed in a recent work (Barchiesi, Kang, Le, Morini, Ponsiglione, SIAM MMS 2010 ) as a variant of the standard perimeter penalization for the denoising of nonsmooth curves.

To deal with the degeneracies, we first give an abstract existence and uniqueness result for viscosity solutions of non-local degenerate Hamiltonians, satisfying suitable continuity assumption with respect to Kuratowsky convergence of the level sets. This abstract setting applies to an approximated flow. Then, by the method of minimizing movements, we also build an ``exact’’ curvature flow. Recent advances towards a better understanding of this flow will be mentioned. This is a joint work with M. Morini (Parma) and M. Ponsiglione (Roma I).