Memoirs of the Graduate School of Science and Engineering, Shimane University. Series B, Mathematics

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Memoirs of the Graduate School of Science and Engineering, Shimane University. Series B, Mathematics 48
2015-03 発行

1-harmonic functions on a network

Kurata, Hisayasu
Yamasaki, Maretsugu
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Description
A minimizer of the Dirichlet norm of order 1 is called a 1-harmonic function. The aim of this paper is a research of properties of 1-harmonic functions on a network. First we consider the 1-Dirichlet space and show that every network is of 1-hyperbolic type and that the ideal boundary coincides with the 1-harmonic boundary. Next we introduce the notion of 1-harmonic functions and that of strongly 1-harmonic functions. We discuss the Dirichlet problem and the maximum principle with respect to (strongly) 1-harmonic functions.