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

Abstract

Let Ω be a sufficiently regular bounded connected open subset of such that 0 ∈ Ω and that is connected. Then we take q11, … ,qnn ∈ ]0,+ ∞ [and . If ε is a small positive number, then we define the periodically perforated domain , where {e1, … ,en} is the canonical basis of . For ε small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set . 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. Copyright © 2011 John Wiley & Sons, Ltd.