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 $$K_{\infty }^G:=\left\{u\in L^2(V,{\nu }_G):|u(y)-u(x)|\le 1ifx\sim y\right\}$$ and the set $$K_{\infty }^w:=\left\{u\in L^2(V,{\nu }_G):|u(y)-u(x)|\le \sqrt{w_{xy}}ifx\sim y\right\}$$ 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 $K_{\infty }^G$ or $K_{\infty }^w$ 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.