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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Bibliographic Details
Published in:Calculus of variations and partial differential equations Vol. 51; no. 1-2; pp. 343 - 362
Main Authors: Hua, Bobo, Keller, Matthias
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2014
Springer Nature B.V
Subjects:
Analysis
Calculus of variations
Calculus of Variations and Optimal Control; Optimization
Control
Criteria
Estimates
Generators
Graph algorithms
Graphs
Harmonic analysis
Harmonic functions
Harmonics
Laplace transforms
Mathematical and Computational Physics
Mathematical functions
Mathematics
Mathematics and Statistics
Systems Theory
Texts
Theorems
Theoretical
58E20
31C05
ISSN:0944-2669, 1432-0835
Online Access:Get full text
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Summary: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.
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ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-013-0677-6