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Mathematics of the USSR-Sbornik, 1986, Volume 55, Issue 2, Pages 493–509
DOI: https://doi.org/10.1070/SM1986v055n02ABEH003017
(Mi sm2011)
 

This article is cited in 5 scientific papers (total in 5 papers)

Second-order elliptic equations on graphs

A. B. Merkov
References:
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
Bibliographic databases:
UDC: 517.95
MSC: Primary 35J15; Secondary 05C10
Language: English
Original paper language: Russian
Citation: A. B. Merkov, “Second-order elliptic equations on graphs”, Math. USSR-Sb., 55:2 (1986), 493–509
Citation in format AMSBIB
\Bibitem{Mer85}
\by A.~B.~Merkov
\paper Second-order elliptic equations on graphs
\jour Math. USSR-Sb.
\yr 1986
\vol 55
\issue 2
\pages 493--509
\mathnet{http://mi.mathnet.ru//eng/sm2011}
\crossref{https://doi.org/10.1070/SM1986v055n02ABEH003017}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=806514}
\zmath{https://zbmath.org/?q=an:0657.35044|0583.35031}
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  • https://doi.org/10.1070/SM1986v055n02ABEH003017
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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