  We revisit the homogenization problem for the Poisson equation in
periodically perforated domains with zero Neumann data at the boundary of the
holes and prescribed Dirichlet data at the outer boundary. It is known that, if
the periodicity of the holes goes to zero but their volume fraction remains
fixed and positive, the limit problem is a Dirichlet boundary value problem
posed in the domain without the holes, and the effective diffusion coefficients
are non-trivially modified; if that volume fraction goes to zero instead, i.e.
the holes are dilute, the effective operator remains the Laplacian (that is,
unmodified). Our main results contain the study of a "continuity" in those
effective models with respect to the volume fraction of the holes and some new
convergence rates for homogenization in the dilute setting. Our method explores
the classical two-scale expansion ansatz and relies on asymptotic analysis of
the rescaled cell problems using layer potential theory.
