# Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector 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 $$\varepsilon$$ ε for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $$\varepsilon$$ ε tends to zero is described not only by asymptotic series in powers of $$\varepsilon$$ ε , 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 $$\omega$$ ω . Then in addition to the scale $$\varepsilon$$ ε there appears the scale $$\eta =\varepsilon ^{\pi /\omega }$$ η = ε π / ω . We prove that when $$\pi /\omega$$ π / ω 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 $$\omega$$ ω (characterized by Diophantine approximation properties), for which real analyticity in the two variables $$\varepsilon$$ ε and $$\eta$$ η holds and the power series converge unconditionally. When $$\pi /\omega$$ π / ω is rational, the series are unconditionally convergent, but contain terms in $$\log \varepsilon$$ log ε . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

# Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

, Volume 88 (3) – May 18, 2017
49 pages

/lp/springer_journal/converging-expansions-for-lipschitz-self-similar-perforations-of-a-6i2iTnkMEC
Publisher
Springer International Publishing
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-017-2377-7
Publisher site
See Article on Publisher Site

### 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 $$\varepsilon$$ ε for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as $$\varepsilon$$ ε tends to zero is described not only by asymptotic series in powers of $$\varepsilon$$ ε , 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 $$\omega$$ ω . Then in addition to the scale $$\varepsilon$$ ε there appears the scale $$\eta =\varepsilon ^{\pi /\omega }$$ η = ε π / ω . We prove that when $$\pi /\omega$$ π / ω 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 $$\omega$$ ω (characterized by Diophantine approximation properties), for which real analyticity in the two variables $$\varepsilon$$ ε and $$\eta$$ η holds and the power series converge unconditionally. When $$\pi /\omega$$ π / ω is rational, the series are unconditionally convergent, but contain terms in $$\log \varepsilon$$ log ε .

### Journal

Integral Equations and Operator TheorySpringer Journals

Published: May 18, 2017

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