Degenerate scale problem for the Laplace equation in the multiply connected region with outer elliptic boundary

This paper investigates the degenerate scale problem for the Laplace equation in a multiply connected region with an outer elliptic boundary. Inside the elliptic boundary, there are many voids with arbitrary configurations. The problem is studied on the relevant homogenous boundary integral equation. The suggested solution is derived from a solution of a relevant problem. It is found that the degenerate scale and the eigen-function along the elliptic boundary in the problem is the same as in the case of a single elliptic contour without voids, or the involved voids have no influence on the degenerate scale. The present study mainly depends on the integrations of two integrals, which can be integrated in closed form.