The limit as p → ∞ in a nonlocal p Laplacian evolution equation: a nonlocal approximation of a model for sandpiles (original) (raw)

2009, Calculus of Variations and Partial Differential Equations

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=uinL2(mathbbRN),:,∣u(x)−u(y)∣le1,mboxwhenx−yinrmsupp(J){K = \{ u \in L^2(\mathbb{R}^N) \, : \, | u(x) - u(y)| \le 1, \mbox{ when } x-y \in {\rm supp} (J)\}}K=uinL2(mathbbRN),:,∣u(x)−u(y)∣le1,mboxwhenx−yinrmsupp(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 = Δp 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.