Title: Converging expansions for Lipschitz self-similar perforations of a plane sector,
Abstract: In contrast with the well-known methods of matching asymptotics and
multiscale (or compound) asymptotics, the " functional analytic approach " of
Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of
expansions around interior small holes of size $\epsilon$ for solutions of
elliptic boundary value problems. Using the method of layer potentials, the
asymptotic behavior of the solution as $\epsilon$ tends to zero is described
not only by asymptotic series in powers of $\epsilon$, but by convergent power
series. Here we use this method to investigate the Dirichlet problem for the
Laplace operator where holes are collapsing at a polygonal corner of opening
$\omega$. Then in addition to the scale $\epsilon$ there appears the scale
$\eta = \epsilon^{\pi/\omega}$. We prove that when $\pi/\omega$ is irrational,
the solution of the Dirichlet problem is given by convergent series in powers
of these two small parameters. Due to interference of the two scales, this
convergence is obtained, in full generality, by grouping together integer
powers of the two scales that are very close to each other. Nevertheless, there
exists a dense subset of openings $\omega$ (characterized by Diophantine
approximation properties), for which real analyticity in the two variables
$\epsilon$ and $\eta$ holds and the power series converge unconditionally. When
$\pi/\omega$ is rational, the series are unconditionally convergent, but
contain terms in log $\epsilon$.