Real analytic families of harmonic functions in a planar domain with a small hole

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ϵ and we denote by uϵ the corresponding solution. The behavior of uϵ for ϵ small and positive can be described in terms of real analytic functions of two variables evaluated at (ϵ,1/log⁡ϵ). We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by uϵ for ϵ small and describe uϵ by real analytic functions of ϵ. Then it is natural to ask what happens when ϵ is negative. The case of boundary data depending on ϵ is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.