Affiliation:
1. Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze, Ed. 8, 90128 Palermo, Italy
2. Dipartimento di Matematica “Tullio Levi Civita,” Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
3. 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 / ϵ n − 2 . In case of dimension n = 2, we have to add log ϵ times the integral of f / 2 π.
Reference39 articles.
1. H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162, Springer, New York, 2007.
2. Effective parameters of elastic composites;Ammari;Indiana Univ. Math. J.,2006
3. Boundary layer techniques for deriving the effective properties of composite materials;Ammari;Asymptot. Anal.,2005
4. Converging expansions for Lipschitz self-similar perforations of a plane sector;Costabel;Integral Equations Oper. Theory,2017
5. Mapping properties of weakly singular periodic volume potentials in Roumieu classes;Dalla Riva;J. Integral Equations Appl.,2020