  Let $\Omega$ be a sufficiently regular bounded open connected subset of
$\mathbb{R}^n$ such that $0 \in \Omega$ and that $\mathbb{R}^n \setminus
\mathrm{cl}\Omega$ is connected. Then we take $q_{11},..., q_{nn}\in
]0,+\infty[$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $\epsilon$ is a
small positive number, then we define the periodically perforated domain
$\mathbb{S}[\Omega_\epsilon]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in
\mathbb{Z}^n}\mathrm{cl}\bigl(p+\epsilon \Omega +\sum_{j=1}^n
(q_{jj}z_j)e_j\bigr)$, where $\{e_1,...,e_n\}$ is the canonical basis of
$\mathbb{R}^n$. For $\epsilon$ small and positive, we introduce a particular
Dirichlet problem for the Laplace operator in the set
$\mathbb{S}[\Omega_\epsilon]^{-}$. Namely, we consider a Dirichlet condition on
the boundary of the set $p+\epsilon \Omega$, 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
$\epsilon$ and of the Dirichlet datum on $p+\epsilon \partial \Omega$, around a
degenerate pair with $\epsilon=0$.
