## Abstract

Let Ω be a sufficiently regular bounded connected open subset of R ^{n} such that 0 ∈ Ω and that R ^{n}\clΩ is connected. Then we take q _{11}, ⋯ ,q _{nn} ∈ ]0,+ ∞ [and p∈Q≡∏ _{j=1} ^{n}]0,q _{jj}[. If ε is a small positive number, then we define the periodically perforated domain S[Ω _{ε}]-≡R ^{n}\ ∪ _{z∈Zn/}cl(p+εΩ+∑ _{j=1} ^{n}(q _{jj}z _{j})e _{j}, where {e _{1}, ⋯ ,e _{n}} is the canonical basis of R ^{n}. For ε small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set S[Ωε]-. Namely, we consider a Dirichlet condition on the boundary of the set p + εΩ, together with a periodicity condition. Then we show real analytic continuation properties of the solution and of the corresponding energy integral as functionals of the pair of ε and of the Dirichlet datum on p + ε∂Ω, around a degenerate pair with ε = 0.

Original language | English |
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Pages (from-to) | 334-349 |

Number of pages | 16 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 35 |

Issue number | 3 |

Early online date | 30 Dec 2011 |

DOIs | |

Publication status | Published - 01 Feb 2012 |

## Keywords

- Boundary value problems for second-order elliptic equations
- integral representations, integral operators, integral equations methods
- Laplace operator
- periodically perforated domain
- real analytic continuation in Banach space
- singularly perturbed domain