Quasi-Symmetricity for the Nonlinear Green Function on a Tree

Abstract

It is well known that the linear Green function is symmetric as a two variable function. However it is not known whether the nonlinear Green function is symmetric or not. Moreover it is not known even whether that is quasi-symmetric or not, which means that the symmetric ratio of the nonlinear Green function is bounded or not. In this article we show that, for every tree, there exists a resistance such that the nonlinear Green function is quasi-symmetric; also we show that, for every tree, there exists a resistance such that the nonlinear Green function is not quasi-symmetric.