Fast/Slow Diffusion and Growing Sandpiles

We study a coupled system of ODE (introduced by the first author inSIAM J. Appl. Math.22(1972) 437–458) for the heights of growing, interacting sand cones. We show that these ODE correspond to the evolution inL2generated by the subdifferential of the convex functional which vanishes on functions whose gradient has length less than or equal to one and is infinity otherwise. Additionally we explain how the ODE arise from evolutions governed by thep-Laplacian in the “infinitely fast/infinitely slow” diffusion limit asp→∞.