Some potential theoretic results on an infinite network. (English) Zbl 1128.31002

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 353-362 (2006).
The greatest harmonic minorant of a superharmonic function is determined as the limit of a sequence of solutions for discrete Dirichlet problems on finite subnetworks. Without using the Green kernel explicitly, a positive superharmonic function is decomposed uniquely as a sum of a potential and a harmonic function. The infimum of a left directed family of harmonic functions is shown to be either \(-\infty\) or harmonic. As applications, the authors study the reduced functions and their properties. Finally, the existence of the Green kernel with the aid of the above reduced functions is shown.
For the entire collection see [Zbl 1102.31001].


31C20 Discrete potential theory
31C05 Harmonic, subharmonic, superharmonic functions on other spaces