Discrete Partial Differential Equations on Graphs and Applications of Ricci Curvature

Abstract

In this paper, we mainly discuss how partial differential equations can be modeled in a graph setting. Most discussions are based on previous studies from the references. In Section 1, we first introduce graph theoretic notions to establish discrete analogues of calculus on graphs. In particular, the operator of our main interest is the discrete Laplace operator. In Section 2, we study discrete versions of linear PDEs such as the Laplace equation, the heat equation, and the wave equation. We also give their properties and show uniqueness via the energy method. In Section 3, we provide a generalization of lower Ricci curvature bound in the discrete framework due to Bakry and Emery. With this, we can get an estimate for the eigenvalue of the Laplace operator on graphs. In Section 4, using the results from Section 3, we investigate various gradient estimates for positive solutions to the heat equation such as Li-Yau estimate and Hamilton estimate on graphs. As an application of the Li-Yau estimate, a Harnack type inequality will be proved. In Section 5, we show the L1-contraction, uniqueness and existence of solutions to the Porous medium equation and discuss the way of interpreting discrete versions of them on graphs.