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 $\epsilon=0$.
Categories: math.AP