Solving Boundary Value Problems on Networks Using Equilibrium Measures

The purpose of this paper is to construct solutions of self-adjoint boundary value problems on finite networks. To this end, we obtain explicit expressions of the Green functions for all different boundary value problems. The method consists of reducing each boundary value problem either to a Dirichlet problem or to a Poisson equation on a new network closely related with the former boundary value problem. In this process we also get an explicit expression of the Poisson kernel for the Dirichlet problem. In all cases, we express the Green function in terms of equilibrium measures solely, which can be obtained as the unique solution of linear programming problems. In particular, we get analytic expressions of the Green function for the following problems: the Poisson equation on a distance-regular graph, the Dirichlet problem on an infinite distance-regular graph, and the Neumann problem on a ball of an homogeneous tree.