Harmonic Functions on Metric Graphs Under the Anti-Kirchhoff Law
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When does an infinite metric graph allow nonconstant bounded harmonic functions under the anti-Kirchhoff transition law? We give a complete answer to this question in the cases where Liouville’s theorem holds, for trees, for graphs with finitely many essential ramification nodes and for generalized lattices. It turns out that the occurrence of nonconstant bounded harmonic functions under the anti-Kirchhoff law differs strongly from the one under the classical continuity condition combined with the Kirchhoff incident flow law.
KeywordsHarmonic functions Liouville’s theorem infinite graphs metric graphs quantum graphs anti-Kirchhoff law generalized lattices
Mathematics Subject Classification35R02 35J25 34B45 05C50 05C10 05C63 31C05
Joachim von Below is grateful to the research group GREDPA at UPC Barcelona for the invitation in 2018. José A. Lubary is grateful to the LMPA Joseph Liouville at ULCO in Calais for the invitation in 2018. The authors are indebted to the anonymous referee for valuable remarks.