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THE $p$-HARMONIC BOUNDARY AND ${D}_{p} $-MASSIVE SUBSETS OF A GRAPH OF BOUNDED DEGREE

Part of: Other generalizations Graph theory Markov processes

Published online by Cambridge University Press:  12 June 2013

MICHAEL J. PULS
MICHAEL J. PULS*
Affiliation:
Department of Mathematics, John Jay College-CUNY, 524 West 59th Street, New York, NY 10019, USA
*
[email protected]
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Abstract

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Let $p$ be a real number greater than one and let $\Gamma $ be a graph of bounded degree. We investigate links between the $p$-harmonic boundary of $\Gamma $ and the ${D}_{p} $-massive subsets of $\Gamma $. In particular, if there are $n$ pairwise disjoint ${D}_{p} $-massive subsets of $\Gamma $, then the $p$-harmonic boundary of $\Gamma $ consists of at least $n$ elements. We show that the converse of this statement is also true.

Keywords

p-harmonic boundaryDp-massive setp-harmonic functionasymptotically constant functionsextreme points of a path

MSC classification

Secondary: 31C20: Discrete potential theory and numerical methods 05C38: Paths and cycles 31C45: Other generalizations (nonlinear potential theory, etc.) 60J50: Boundary theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 149 - 158
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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