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 (Munich) 28:63–93, 2008) allows to prove convergence of expansions around interior small holes of size $\varepsilon$ for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $\varepsilon$ tends to zero is described not only by asymptotic series in powers of $\varepsilon$, 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 $\varepsilon$ there appears the scale $\eta =\varepsilon ^{\pi /\omega }$. 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 $\varepsilon$ and $\eta$ holds and the power series converge unconditionally. When $\pi /\omega$ is rational, the series are unconditionally convergent, but contain terms in $\log \varepsilon$.