Variational evolution problems and nonlocal geometric motion

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 pr oblems and nonlocal geometric curvature motion: The distance functions to s urfaces moving according to certain nonlocal geometric laws are solutions o f the variational evolution problems. Thus we do the first step of the proo f of heuristics developed in earlier works. The main techniques we use are differential-equation methods in the Monge-Kantorovich theory.