Strong Localized Perturbations and the Biharmonic Problem

Here Ω is the unit disk centered at the origin, containing a small hole of radius ε centered at x = 0, i.e. Ωε = {x | |x| ≤ ε}. Calculate the exact solution, and from it determine an approximation to the solution in the outer region |x| ≫ O(ε). Can you re-derive this result from singular perturbation theory in the limit ε → 0? (Hint: the leading-order outer problem for Case I is different from what you might expect). Solution We first find the exact solution of (1.1) and then expand it for ε → 0. Since the radially symmetric solutions to (1.1 a) are linear combinations of {r, r log r, log r, 1}, we can write the solution to (1.1 a), which satisfies (1.1 c), as