## Abstract

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ε and we denote by u_{ε} the corresponding solution. The behavior of u_{ε} 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 u_{ε} for ε small and describe u_{ε} 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.

Original language | English |
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Pages (from-to) | 37-55 |

Number of pages | 19 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 422 |

Issue number | 1 |

Early online date | 22 Aug 2014 |

DOIs | |

Publication status | Published - 01 Feb 2015 |

## Keywords

- Harmonic functions
- Real analytic continuation in Banach space
- Singularly perturbed perforated planar domains