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, then there exists an analytic mapping f from U onto the open unit disk D. This mapping is known as a Riemann mapping.
Since the composition of a harmonic and analytic function is harmonic, the Riemann mapping provides a bijection between harmonic functions defined on the set U and on the disc D. Therefore, one can transfer a solution of a Dirichlet problem on the set D to the set U.
be a Riemann mapping 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 of the exercise above is true, see also [Ca].
Exercise (**). Let G be a network, with the Kirchhoff matrix
For a positive vector x, let Dx denote the diagonal matrix with the vector x on its diagonal. That is Dx1=x. Find a new conductivity on the network G, such that
(Hint). , where
is the solution of the Dirichlet 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(!)
with multiplicity 2 and 0 w/multiplicity 1.