# On 2D Inverse Problems/Riemann mapping theorem

In complex analysis, the **Riemann mapping theorem** states that for every non-empty simply connected open subset *U* of the complex plane *C*, which is not all of the complex plane, there exists a 1-to-1 conformal **Riemann map** from *U* onto the open unit disk *D*. Since the composition of a harmonic and analytic function is harmonic, the Riemann map provides a 1-to-1 correspondence b/w harmonic functions defined on the set *U* and on the disc *D*. Therefore, one can transfer a solution of a Dirichlet boundary problem on the set *D* to the set *U*.

Let be a Riemann map for the region *U*, then the kernel of the Dirichlet-to-Neumann map for the region *U* can be expressed in terms of the Dirichlet-to-Neumann map for the disc.

**Exercise (*).**Proof that,

off the diagonal.

It is a remarkable fact that a discrete/network version of the statement above is true, see also [Ca].

**Exercise (**).**Let*G*be a network w/the Kirchhoff matrix

Find a new conductivity on the network *G*, such that

(Hint). , where

is the solution of the Dirichlet boundary problem and

Compare to the continuous case.

**Exercise (*)**

Prove that the **Cayley transform**

is a Riemann mapping of the complex right half-plane *C*^{+} onto the unit disc *D*

**Exercise (**)**Use statements above to derive the formula for the kernel of the Dirichlet-to-Neumann map for the unit disc*D*.

Note, that the formula can also be derived by taking the radial derivative of the **Poisson kernel** for solving a Dirichlet problem on the disc.

In order to solve a continuous inverse problem by data discretization, one can define a Dirichlet-to-Neumann (DN) matrix by uniform sampling of the kernel off the diagonal, and defining the diagonal entries by the fact that rows and columns of a DN matrix sum up to zero. This leads to the following definition of the matrix in the case of the unit disc:

where *n* is a natural number and *k,l = 1,2, ... 2n+1*.

**Exercise (***)**. Prove that the eigenvalues of the matrix above are natural numbers(!)

w/multiplicity *2* and *0* w/multiplicity *1*.