Harmonic functions on locally finite networks |
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Authors: | Joachim von Below José A Lubary |
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Institution: | 1. LMPA Joseph Liouville, EA 2597, Université du Littoral C?te d’Opale, 50, rue F. Buisson, B.P. 699, F-62228, Calais Cedex, France 2. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028, Barcelona, Spain
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Abstract: | Are there nonconstant bounded harmonic functions on an infinite locally finite network under natural transition conditions as continuity at the ramification nodes and classical Kirchhoff conditions at all vertices? We present sufficient criteria for such a network to be a Liouville space, while we show that a large class of infinite trees admit infinitely many linearly independent bounded harmonic functions. Finally, we show that the standard unit cube grid graphs and some of Kepler’s plane tiling graphs are Liouville spaces. |
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