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 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 $\Lambda$ satisfies a set of inequalities of the form $(-1)^{\frac{n(n+1)}{2}}\det K(x_i,y_j)>0$. 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.