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We address in this paper the study of a geometric evolution, corresponding toa curvature which is non-local and singular at the origin. The curvaturerepresents the first variation of the energy recently proposed as a variant ofthe standard perimeter penalization for the denoising of nonsmooth curves.  To deal with such degeneracies, we first give an abstract existence anduniqueness result for viscosity solutions of non-local degenerate Hamiltonians,satisfying suitable continuity assumption with respect to Kuratowskyconvergence of the level sets. This abstract setting applies to an approximatedflow. Then, by the method of minimizing movements, we also build an "exact"curvature flow, and we illustrate some examples, comparing the results with thestandard mean curvature flow.

A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation