Fast/slow diffusion and collapsing sandpiles

We regard the limit as p →∞ of the flow governed by the p-Laplacian as providing a simplistic model for the “collapse of an initially unstable sandpile.” Upon rescaling to stretch out the initial layer we obtain some simple dynamics and provide fairly explicit solutions in certain cases involving nonlocal geometric curvature motion. In particular we note that such models entail “instantaneous” mass transfer governed by Monge-Kantorovich theory.