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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.
UDC:
517.95
MSC: Primary
35J15; Secondary
05C10 Received: 07.09.1984
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