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 Lambda for the electric al conductivity equation on a simply connected plane region has an alt ernating property, which may be considered as a generalized maximum pr inciple. Using this property, we will prove that the kernel, K, of Lam bda satisfies a set of inequalities of the form (-1)(n(n+1)/2) det K(x (i), y(j)) > 0. We will show that these inequalities imply Hopf's lemm a for the conductivity equation. We will also show that these inequali ties imply the alternating property of a kernel.