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

Let $Omega$ be a sufficiently regular bounded open connected subset of $mathbbR^n$ such that $0 in Omega$ and that $mathbbR^n setminus mathrmclOmega$ is connected. Then we take $(q_11,dots, q_nn)in ]0,+infty[^n$ and $p in Qequiv prod_j=1^n]0,q_jj[$. If $epsilon$ is a small positive number, then we define the periodically perforated domain $mathbbS[Omega_p,epsilon]^- equiv mathbbR^nsetminus cup_z in mathbbZ^nmathrmcligl(p+epsilon Omega +sum_j=1^n (q_jjz_j)e_jigr)$, where $e_1,dots,e_n$ is the canonical basis of $mathbbR^n$. For $epsilon$ small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set $mathbbS[Omega_p,epsilon]^-$. 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 as a function of $epsilon$, of the Dirichlet datum on $p+epsilon partial Omega$, and of the Poisson datum, around a degenerate triple with $epsilon=0$.