On 2D Inverse Problems/Kernel of Dirichlet-to-Neumann map

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The continuous analog of the matrix representation of a Dirichlet-to-Neumann operator for a domain  is its kernel. It is a distribution defined on the Cartesian product of the boundary of the domain w/itself, such that if  

then where and parametrize the boundary w/arclength measure.

For the case of the half-plane w/constant unit conductivity the kernel can be calculated explicitly. It's a convolution, because the domain in consideration is shift invariant:

where k is a distribution on a line. Therefore, the calculation reduces to solving the Dirichlet problem for a -function at the origin and taking outward derivative at the boundary line.

Dirichlet problem for a half-plane

Exercise (**). Complete the calculation of the kernel K for the half-plane to show that off the diagonal.

Exercise (*). Prove that for rotation invariant domain (disc w/ conductivity depending only on radius) the kernel of Dirichlet-to-Neumann map is a convolution.

The Hilbert transform gives a correspondence between boundary values of a harmonic function and its harmonic conjugate. where is an analytic function in the domain.

For the case of the complex upper half-plane the Hilbert transform is given by the following formula:

Exercise (*). Differentiate under the integral sign the formula above to obtain the kernel representation for the Dirichlet-to-Neumann operator for the half plane.

To define discrete Hilbert transform for a planar network, the network together w/its dual is needed.