Two models forsandpile growth in weighted graphs

Abstract

In this paper we study $\infty$-Laplacian type diffusion equations in weighted graphs obtained as limit as $p\to \infty$ to two types of $p$-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set $$K^G_{\infty}:= \left\{ u \in L^2(V, \nu_G) \ : \ \vert u(y) - u(x) \vert \leq 1 \ \ \hbox{if} \ \ x \sim y \right\}$$ and the set $$K^w_{\infty}:= \left\{ u \in L^2(V, \nu_G) \ : \ \vert u(y) - u(x) \vert \leq \sqrt{w_{xy}} \ \ \hbox{if} \ \ x \sim y \right\}$$ as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets $K^G_{\infty}$ or $K^w_{\infty}$ by means of an abstract result given in~\cite{BEG}. We give an interpretation of the limit problems in terms of Monge-Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing.