A singularly perturbed Dirichlet problem for the Laplace operator 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 $q_11,dots, q_nnin ]0,+infty[$ 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 $mathbbS[Omega_epsilon]^- equiv mathbbR^nsetminus cup_z in mathbbZ^nmathrmcligl(p+epsilon Omega +sum_j=1^n (q_jjz_j)e_jigr)$, 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 Laplace operator in the set $mathbbS[Omega_epsilon]^-$. 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 $epsilon=0$.

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=1n]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=1n(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. © 2011 John Wiley & Sons, Ltd.