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If $\\epsilon$ is a small positive real number, then we define the periodically perforated domain $T(\\epsilon) \\equiv \\mathbb{R}^n\\setminus \\cup_{z \\in \\mathbb{Z}^n}\\cl(w+\\epsilon \\Omega +z)$. For each small positive $\\epsilon$, we introduce a particular Dirichlet problem for the Laplace operator in the set $T(\\epsilon)$. 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