THE LIMIT AS p →∞ IN A NONLOCAL p−LAPLACIAN EVOLUTION EQUATION. A NONLOCAL APPROXIMATION OF A MODEL FOR SANDPILES

In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p →∞ to the nonlocal analogous to the p−Laplacian evolution, ut(t, x) = ∫ RN J(x− y)|u(t, y)− u(t, x)|p−2(u(t, y)− u(t, x)) dy. We prove existence and uniqueness of a limit solution that verifies an equation governed by the subdifferetial of a convex energy functional associated to the indicator function of the set K = {u ∈ L(R ) : |u(x)−u(y)| ≤ 1, when x− y ∈ supp(J)}. We also find some explicit examples of solutions to the limit equation. If the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞(0, T ; L(Ω)) to the limit solution of the local evolutions of the p−laplacian, vt = ∆pv. This last limit problem has been proposed as a model to describe the formation of a sandpile. Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large. Finally, we give an interpretation of the limit problem in terms of Monge-Kantorovich mass transport theory.