Degenerate scale for the Laplace problem in the half-plane; Approximate logarithmic capacity for two distant boundaries

We study the problem of finding a degenerate scale for Laplace equation in a half-plane. It is shown that if the boundary condition on the line bounding the half-plane is of Dirichlet type, there is no degenerate scale. In the case of a boundary condition of Neumann type, there is a degenerate scale, which is shown to be the same as the one for the symmetrized contour with respect to the boundary line in the full plane. We show next a formula for obtaining the degenerate scale of a domain made of two parts, when the components are far from each other, which allows to obtain the degenerate scale for the symmetrized contour. Finally, we give some examples of evaluation of the degenerate scale both by an approximate formula and by a numeric evaluation using integral methods. These evaluations show that the approximate solution is still valid for small values of the distance between symmetrized contours.