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 $epsilon$ and we denote by $u_epsilon$ the corresponding solution. The behavior of $u_epsilon$ for $epsilon$ small and positive can be described in terms of real analytic functions of two variables evaluated at $(epsilon,1/logepsilon)$. We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by $u_epsilon$ for $epsilon$ small and describe $u_epsilon$ by real analytic functions of $epsilon$. Then it is natural to ask what happens when $epsilon$ is negative. The case of boundary data depending on $epsilon$ 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.

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.