The L2 strong maximum principle on arbitrary countable networks

Details The L2 strong maximum principle on arbitrary countable networks The L2 strong maximum principle on arbitrary countable networks

2009-02-02

arxiv.org/abs/0902.0251v2

Mathematics

Analysis of PDEs

14 pages; accepted in "Linear algebra and its applications"

Scientific paper

10.1016/j.laa.2011.03.001

We study the strong maximum principle for the heat equation associated with the Dirichlet form on countable networks. We start by analyzing the boundedness properties of the incidence operators on a countable network. Subsequently, we prove that the strong maximum principle is equivalent to the underlying graph being connected after deletion of the nodes with infinite degree. Using this result, we prove that the number of connected components of the graph with respect to the heat flow equals the number of maximal invariant ideals of the adjacency matrix.

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