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 Algebra i Analiz, 2007, Volume 19, Issue 2, Pages 183–225 (Mi aa109)

This article is cited in 9 scientific papers (total in 9 papers)

Research Papers

Dirichlet problem in an angular domain with rapidly oscillating boundary: Modeling of the problem and asymptotics of the solution

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Peterburg

Abstract: Leading asymptotic terms are constructed and justified for the solution of the Dirichlet problem corresponding to the Poisson equation in an angular domain with rapidly oscillating boundary. In addition to an exponential boundary layer near the entire boundary, a power-law boundary layer arises, which is localized in the vicinity of the corner point. Modeling of the problem in a singularly perturbed domain is studied; this amounts to finding a boundary-value problem in a simpler domain whose solution approximates that of the initial problem with advanced precision, namely, yields a two-term asymptotic expression. The way of modeling depends on the opening $\alpha$ of the angle at the corner point; the cases where $\alpha<\pi$, $\alpha\in(\pi,2\pi)$, and $\alpha=2\pi$ are treated differently, and some of them require the techniques of selfadjoint extensions of differential operators.

Keywords: Dirichlet problem, oscillating boundary, corner point, asymptotics, selfadjoint extension.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:2, 297–326

Bibliographic databases:

MSC: 35B40, 35J25
Received: 10.10.2006

Citation: S. A. Nazarov, “Dirichlet problem in an angular domain with rapidly oscillating boundary: Modeling of the problem and asymptotics of the solution”, Algebra i Analiz, 19:2 (2007), 183–225; St. Petersburg Math. J., 19:2 (2008), 297–326

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. A. Nazarov, “Asymptotic modeling of a problem with contrasting stiffness”, J. Math. Sci. (N. Y.), 167:5 (2010), 692–712
2. V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, St. Petersburg Math. J., 22:6 (2011), 941–983
3. Borisov D. Cardone G. Faella L. Perugia C., “Uniform Resolvent Convergence for Strip with Fast Oscillating Boundary”, J. Differ. Equ., 255:12 (2013), 4378–4402
4. Borisov D. Cardone G. Durante T., “Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve”, Proc. R. Soc. Edinb. Sect. A-Math., 146:6 (2016), 1115–1158
5. Hewett D.P., Hewitt I.J., “Homogenized boundary conditions and resonance effects in Faraday cages”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 472:2189 (2016), 20160062
6. Delourme B., Schmidt K., Semin A., “On the homogenization of thin perforated walls of finite length”, Asymptotic Anal., 97:3-4 (2016), 211–264
7. Cardone G., “Waveguides With Fast Oscillating Boundary”, Nanosyst.-Phys. Chem. Math., 8:2 (2017), 160–165
8. Semin A., Schmidt K., “On the Homogenization of the Acoustic Wave Propagation in Perforated Ducts of Finite Length For An Inviscid and a Viscous Model”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 474:2210 (2018), 20170708
9. Bunoiu R., Cardone G., Nazarov S.A., “Scalar Problems in Junctions of Rods and a Plate II. Self-Adjoint Extensions and Simulation Models”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 52:2 (2018), 481–508
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