|
This article is cited in 5 scientific papers (total in 5 papers)
Second-order elliptic equations on graphs
A. B. Merkov
Abstract:
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.
Received: 07.09.1984
Citation:
A. B. Merkov, “Second-order elliptic equations on graphs”, Mat. Sb. (N.S.), 127(169):4(8) (1985), 502–518; Math. USSR-Sb., 55:2 (1986), 493–509
Linking options:
https://www.mathnet.ru/eng/sm2011https://doi.org/10.1070/SM1986v055n02ABEH003017 https://www.mathnet.ru/eng/sm/v169/i4/p502
|
Statistics & downloads: |
Abstract page: | 442 | Russian version PDF: | 137 | English version PDF: | 6 | References: | 42 |
|