  We consider the Poisson equation with homogeneous Dirichlet conditions in a
family of domains in $R^{n}$ indexed by a small parameter $\epsilon$. The
domains depend on $\epsilon$ only within a ball of radius proportional to
$\epsilon$ and, as $\epsilon$ tends to zero, they converge in a self-similar
way to a domain with a conical boundary singularity. We construct an expansion
of the solution as a series of fractional powers of $\epsilon$, and prove that
it is not just an asymptotic expansion as $\epsilon\to0$, but that, for small
values of $\epsilon$, it converges normally in the Sobolev space $H^{1}$. The
phenomenon that solutions to boundary value problems on singularly perturbed
domains may have convergent expansions is the subject of the Functional
Analytic Approach by Lanza de Cristoforis and his collaborators. This approach
was originally adopted to study small holes shrinking to interior points of a
smooth domain and heavily relies on integral representations obtained through
layer potentials. To relax all regularity assumptions, we forgo boundary layer
potentials and instead exploit expansions in terms of eigenfunctions of the
Laplace-Beltrami operator on the intersection of the cone with the unit sphere.
Our analysis is based on a two-scale cross-cutoff ansatz for the solution.
Specifically, we write the solution as a sum of a function in the slow variable
multiplied by a cutoff function depending on the fast variable, plus a function
in the fast variable multiplied by a cutoff function depending on the slow
variable. While the cutoffs are considered fixed, the two unknown functions are
solutions to a $2\times2$ system of partial differential equations that depend
on $\epsilon$ in a way that can be analyzed in the framework of generalized
power series when the right-hand side of the Poisson equation vanishes in a
neighborhood of the perturbation.
