The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles

Abstract

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,

$$u_t (t,x) = \int_{\mathbb{R}^N} J(x-y)|u(t,y) - u(t,x)|^{p-2}(u(t,y)- u(t,x)) \, dy.$$

We prove exist ence and uniqueness of a limit solution that verifies an equation governed by the subdifferential of a convex energy functional associated to the indicator function of the set ${K = \{ u \in L^2(\mathbb{R}^N) \, : \, | u(x) - u(y)| \le 1, \mbox{ when } x-y \in {\rm supp} (J)\}}$ . 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 ² (Ω)) to the limit solution of the local evolutions of the p-Laplacian, v _t  = Δ _p v. 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.