TITLE: Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain?
QUESTION [11 upvotes]: Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{const}$ on each electrode). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance.
The Dirichlet-to-Neumann map or Poincare–Steklov operator is the map from voltages to total current on electrodes for a given conductivity distribution (and is a linear map according to Ohm's law and Kirchoff's law (and therefore a matrix)).
Given an arbitrary $n\times n$ matrix, is there a way to check if there exist a conductivity distribution which has this matrix as a Dirichlet-to-Neumann map?
I am looking for a computationally fast one hopefully, I don't want to compute the conductivity distribution.
Maybe there are some domain independent restrictions? (I know there are because I've seen them before but I can't find it again). But I was hoping you could improve them when you know the domain. Further I also know that the conductivity is bounded above by some constant. Is there a way to find what restriction on the DtN map this gives?

REPLY [7 votes]: In dimensions 3 and higher, and without any constraints on $\sigma$, one can apparently obtain any symmetric matrix $A = (a_{ij})$ such that $a_{ij} < 0$ when $i \ne j$ and $a_{ii} = -\sum_{j \ne i} a_{ij}$, as suggested in Denis Serre's answer. That answer already explains why these conditions are necessary. To see that they are also sufficient, one can approximate electrodes connected by non-crossing wires of given conductance (in the limiting case $\sigma$ is zero outside the wires, and constant in each wire).
In dimension 2, I believe, there are additional geometric constraints (for example, it is not possible to connect the electrodes with non-crossing wires). I bet this has been studied, but I do not know any references.

REPLY [6 votes]: This question (for two dimensional domains) was answered by Curtis, Ingerman and Morrow, "Circular Graphs and planar Resistor Networks" (1998). Let $a$ be the $n \times n$ response matrix. As already noted, we must have $a_{ij} = a_{ji}$ and $a(1\ 1\ \cdots\ 1)^T=0$. The additional condition is that, if $i_1$, $i_2$, ..., $i_k$ and $j_1$, $j_2$, ..., $j_k$ appear in circular order around the boundary of the network, and $A'$ is the submatrix with rows $i_r$ and columns $j_s$, then $(-1)^k \det A' \geq 0$.