Abstract
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.
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References
von Below, J., Mugnolo, D.: The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions. Linear Algebra Appl. 439, 1792–1814 (2013)
Mugnolo, D.: Vector-valued heat equations and networks with coupled dynamic boundary conditions. Adv. Differ. Equ. 15, 1125–1160 (2010)
von Below, J., Lubary, J.A.: Harmonic functions on locally finite networks. Results Math. 45, 1–20 (2004)
Biggs, N.L.: Algebraic Graph Theory. Cambridge Tracts Mathematics, vol. 67, 1st edn. Cambridge University Press (1967)
Biggs, N.L.: Algebraic Graph Theory. Cambridge Tracts Mathematics, vol. 67, 2nd edn. Cambridge University Press (1993)
Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21, 209–234 (1989)
Wilson, R.J.: Introduction to Graph Theory. Oliver & Boyd, Edinburgh (1972)
von Below, J., Lubary, J.A.: The eigenvalues of the Laplacian on locally finite networks. Results Math. 47, 199–225 (2005)
von Below, J.: A characteristic equation associated with an eigenvalue problem on \({C}^2\)-networks. Linear Algebra Appl. 71, 309–325 (1985)
von Below, J., Lubary, J.A., Vasseur, B.: Some remarks on the eigenvalue multiplicities of the Laplacian on infinite locally finite trees. Results Math. 63, 1331–1350 (2013)
von Below, J.: The index of a periodic graph. Results Math. 25, 198–223 (1994)
Collatz, L.: Spektren periodischer Graphen. Result. Math. 1, 42–53 (1979)
Woess, W.: Random Walks on Infinite Graphs and Groups, vol. 138. Cambridge University Press, Cambridge (2000)
Acknowledgements
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.
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José A. Lubary was supported by MINECO Grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P and part of the Catalan research group 2017 SGR 1392.
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von Below, J., Lubary, J.A. Harmonic Functions on Metric Graphs Under the Anti-Kirchhoff Law. Results Math 74, 36 (2019). https://doi.org/10.1007/s00025-019-0966-2
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DOI: https://doi.org/10.1007/s00025-019-0966-2
Keywords
- Harmonic functions
- Liouville’s theorem
- infinite graphs
- metric graphs
- quantum graphs
- anti-Kirchhoff law
- generalized lattices