Two models for sandpile growth in weighted graphs

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 $K_{\infty }^G≔\left(u\in L^2(V,{\nu }_G):|u\left(y\right)-u\left(x\right)|\le 1ifx\sim y\right)$ and the set $K_{\infty }^w≔\left(u\in L^2(V,{\nu }_G):|u\left(y\right)-u\left(x\right)|\le \frac{1}{\sqrt{w_{xy}}}ifx\sim y\right)$ 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 Bénilan (2003). 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.