A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities I. The case of an open mid-curve

Asymptotic approximations of voltage potentials in the presence of diametrically small inhomogeneities are well studied. In particular it is known that one may construct approximations that are accurate to any order (in the diameter) uniformly in the conductivity of the inhomogeneity. The corresponding problem for thin inhomogeneities is not so well understood, in particular as concerns uniformity of the approximations. If the conductivity degenerates to 0 or goes to infinity as the width of the inhomogeneity goes to zero, the voltage potential may converge to different limiting solutions, and so the construction of uniform approximations is not straightforward. For the case of thin two dimensional inhomogeneities with closed mid-curves such approximations were constructed and rigorously verified in (Chinese Annals of Mathematics, Series B 38 (2017) 293-344). The analysis relied heavily on the regularity of the approximate solutions. In this two part paper we continue this line of research, by showing that the same approximations remain valid, even when the mid-curve is open, and the corresponding approximate solutions have singularities at the endpoints of the curve.