Harmonic functions of general graph Laplacians

We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an L p Liouville type theorem which is a quantitative integral L p estimate of harmonic functions analogous to Karp’s theorem for Riemannian manifolds. As corollaries we obtain Yau’s L p -L...

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Veröffentlicht in:Calculus of variations and partial differential equations 2014-09, Vol.51 (1-2), p.343-362
Hauptverfasser: Hua, Bobo, Keller, Matthias
Format: Artikel
Sprache:eng
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Zusammenfassung:We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an L p Liouville type theorem which is a quantitative integral L p estimate of harmonic functions analogous to Karp’s theorem for Riemannian manifolds. As corollaries we obtain Yau’s L p -Liouville type theorem on graphs, identify the domain of the generator of the semigroup on L p and get a criterion for recurrence. As a side product, we show an analogue of Yau’s L p Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-013-0677-6