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.
|Number of pages||19|
|Journal||Journal of Mathematical Analysis and Applications|
|Early online date||22 Aug 2014|
|Publication status||Published - 01 Feb 2015|
- Harmonic functions
- Real analytic continuation in Banach space
- Singularly perturbed perforated planar domains