  We consider two variational evolution problems related to Monge-Kantorovich
mass transfer. These problems provide models for collapsing sandpiles and for
compression molding. We prove the following connection between these problems
and nonlocal geometric curvature motion: The distance functions to surfaces
moving according to certain nonlocal geometric laws are solutions of the
variational evolution problems. Thus we do the first step of the proof of
heuristics developed in earlier works. The main techniques we use are
differential equations methods in the Monge-Kantorovich theory.
