Nonlinear Electrical Networks

We consider electrical conductance networks in which the conductors are non-ohmic, so that current is not given by a linear function of voltage. This generalizes the directed current networks of Orion Bawdon. Using convex functions, we give short proofs of the key results of Christianson and Erickson [1]. We show the well-definedness of the Dirichlet-to-Neumann map for arbitrary nonlinear conductance networks. We also consider the dual notion of a nonlinear resistance network, and show that the Neumann-to-Dirichlet map is well-defined for connected graphs. In the process, we demonstrate existence and partial uniqueness results for both problems.