Degenerate scale for 2D Laplace equation with mixed boundary condition and comparison with other conditions on the boundary

It is well known that the 2D Laplace Dirichlet problem has a degenerate scale for which the direct boundary integral equation has several solutions. We study here the case of the mixed boundary condition, mainly for the exterior problem, and show that this problem has also one degenerate scale. The degenerate scale factor is a growing function of the part of the boundary submitted to Neumann condition. Different special cases are then addressed: segment, circle and symmetric problems. Some exact values of the degenerate scale factor are given for equilateral triangle and square. The numerical procedure for determining the degenerate scale factor for mixed BC is described. The comparison is made with other kinds of boundary conditions and the consequence of the choice of Green's function when using the Boundary Element Method is studied.