Second-order elliptic equations on graphs

A. B. Merkov

Sbornik. Mathematics, Tome 55 (1986) no. 2, pp. 493-509

Résumé

The author considers an undirected graph $G$ which, generally speaking, is infinite but has a finite number of edges issuing from each vertex. To each edge $[x,y]$ of the graph there is assigned a positive number $ r_{[x,y]}$ – its “resistance”. A real-valued function $u$ defined on the vertices of $G$ is called elliptic if for each vertex $x\in G$ the following condition holds: $$ Lu(x)=\sum_{[x,y]\in G}\frac{u(y)-u(x)}{r_{[x,y]}}=0. $$ It is shown that under certain conditions on the graph and the resistance of its edges elliptic functions behave like solutions of second-order uniformly elliptic equations of divergence form without lower-order terms on $\mathbf{R}^n$. In particular, analogues of Harnack's inequality and Liouville's theorem hold for them. The concept of a fundamental solution of the operator $L$ is introduced, and some conditions for the existence of a positive fundamental solution of the operator $L$ on the graph $G$ are given. Figures: 1. Bibliography: 2 titles.