Potential Theory on Finite Networks

Abstract

Solving Kirchhoff’s problem in a finite electrical network opens up various possibilities for the study of functions in networks. A method to solve the Dirichlet-Poisson equation in a finite network, by using algebraic techniques involving the inverses of submatrices of the Laplace matrix is given here; the novelty is the Laplace matrix need not be symmetric. The same method is used to explain Equilibrium Principle, Condenser Principle and Balayage. Later, assuming the Laplace matrix is symmetric, the Dirichlet semi-norm is defined and the Dirichlet Principle is proved. This procedure can be modified to develop a potential theory associated with the Schrödinger operators in a finite network. However, another approach based on the properties of Schrödinger superharmonic functions is adopted here in order to bring into focus the potential-theoretic methods that will be used later in the context of infinite networks.