A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach

Paolo Musolino*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Citations (SciVal)

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 jjz 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 languageEnglish
Pages (from-to)334-349
Number of pages16
JournalMathematical Methods in the Applied Sciences
Volume35
Issue number3
Early online date30 Dec 2011
DOIs
Publication statusPublished - 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

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