Singular behavior for a multi-parameter periodic Dirichlet problem

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 I > 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 ( I , φ , g , f ). Our aim is to study how the solution depends on ( I , φ , g , f ), especially when I 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 I 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 ( I , φ , g , f ) multiplied by the factor 1 / I n-2 . In case of dimension n = 2, we have to add log I times the integral of f / 2 I .