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

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 detK(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.