A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach
Паоло Мусоліно · Mathematical Methods in the Applied Sciences · 2011
Let Ω be a sufficiently regular bounded connected open subset of such that 0 ∈ Ω and that is connected. Then we takeq11, … ,qnn ∈ ]0,+ ∞ [and . Ifεis a small positive number, then we define the periodically perforated domain , where {e1, … ,en} is the canonical basis of . Forεsmall 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 setp + 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 ofεand of the Dirichlet datum onp + ε∂Ω, around a degenerate pair withε = 0. Copyright © 2011 John Wiley & Sons, Ltd.