Affiliations: [a] Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze, Ed. 8, 90128 Palermo, Italy | [b] Dipartimento di Matematica “Tullio Levi Civita,” Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy | [c] Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca’ Foscari Venezia, Via Torino 155, 30172 Venezia Mestre, Italy
Abstract: We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number ϵ>0, proportional to the radius of the holes, and a map ϕ, which models the shape of the holes. So, if g denotes the Dirichlet boundary datum and f the Poisson datum, we have a solution for each quadruple (ϵ,ϕ,g,f). Our aim is to study how the solution depends on (ϵ,ϕ,g,f), especially when ϵ is very small and the holes narrow to points. In contrast with previous works, we do not introduce the assumption that f has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for ϵ close to 0. We show that, when the dimension n of the ambient space is greater than or equal to 3, a suitable restriction of the solution can be represented with an analytic map of the quadruple (ϵ,ϕ,g,f) multiplied by the factor 1/ϵn−2. In case of dimension n=2, we have to add logϵ times the integral of f/2π.
Keywords: Dirichlet problem, integral equation method, Poisson equation, periodically perforated domain, singularly perturbed domain, real analytic continuation in Banach spaces
