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.