On a Characterization of the Kernel of the Dirichlet-to-Neumann Map for a Planar Region

Abstract

We will show that the Dirichlet-to-Neumann map Λ for the electrical conductivity equation on a simply connected plane region has an alternating property, which may be considered as a generalized maximum principle. Using this property, we will prove that the kernel, K, of Λ satisfies a set of inequalities of the form (−1) n(n+1) 2 det K(xi, yj) > 0. We will show that these inequalities imply Hopf's lemma for the conductivity equation. We will also show that these inequalities imply the alternating property of a kernel.

Dirichlet-to-Robin maps on finite networks

“…These properties are analogs of those that characterize the Dirichlet-toNeumann maps, see [8] for circular planar networks and [11] for the continuous case. In particular, we prove that the Dirichlet-to-Robin map of a general network is a self-adjoint, positive semi-definite operator, whose kernel is related with the second normal derivative of the Green operator, and it is negative off-diagonal and positive on the diagonal.…”

“…This point of view is important in applications since finite network models arise in finite volume discretizations of the elliptic partial differential equation that model the continuos inverse problem, see [5,6,11]. In this framework, the characterization of the networks that allow the recovery of the conductance is essential.…”

Overdetermined partial boundary value problems on finite networks

“…For instance, Sylvester and Uhlmann treated in [9,18] the uniqueness of solution; Curtis, Ingerman and Morrow have worked on critical circular planar networks conductivity reconstruction [12,13,14,16]; Borcea, Druskin, Guevara and Mamonov have gone into EIT problems in depth and their last works on the subject treat numerical conductivity reconstruction [6,7,8].…”

Discrete Evolution Equations on Networks and a Unique Identifiability of Their Weights

“…Among these studies, solving forward and inverse problems for equations by means of an elliptic operator, called an ω-Laplacian ∆ ω on networks, which can be interpreted as a diffusion equation on graphs modeled by electric networks, have been investigated by a lot of authors, because of their applications to many practical examples such as identification of conductivity or finding perturbation of electric networks. See, for example, [4], [5], [8], [9], [10], [12] and [13].…”