Fast/slow Diffusion and Collapsing Sandpiles

We regard the limit as p ! 1 of the ow 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. In particular we note that such models entail \instanta-neous" mass transfer governed by Monge-Kantorovich theory. 1. Introduction We study in this paper a \fast/slow diiusion" PDE problem, which can be very loosely interpreted as modelling the collapse of a sandpile from an initially unstable connguration.