Discrete Harmonic Functions on Infinite Penny Graphs

Abstract

In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincaré inequality hold, which yields the Harnack inequality for positive harmonic functions. Moreover, we prove that the space of polynomial growth harmonic functions, or ancient solutions of the heat equation, with bounded growth rate has finite dimensional property.

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Acknowledgements

The author is supported by NSFC, no.11831004 and no. 11926313.

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Correspondence to Bobo Hua.

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This is dedicated to Professor Jürgen Jost on the occasion of his 65th birthday.

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Hua, B. Discrete Harmonic Functions on Infinite Penny Graphs. Vietnam J. Math. (2021). https://doi.org/10.1007/s10013-020-00471-7

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Keywords

  • Penny graphs
  • Harmonic functions
  • Quasi-isometry

Mathematics Subject Classification (2010)

  • 05C10
  • 31C05