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 Mat. Sb., 2000, Volume 191, Number 6, Pages 43–68 (Mi msb483)

On the Dirichlet problem for the Helmholtz equation on the plane with boundary conditions on an almost closed curve

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Abstract: In this article the two-dimensional Dirichlet boundary-value problem is considered for the Helmholtz operator with boundary conditions on an almost closed curve $\Gamma_\varepsilon$ where $\varepsilon\ll 1$ is the distance between the end-points of the curve. A complete asymptotic expression is constructed for a pole of the analytic continuation of the Green's function of this problem as the pole converges to a simple eigenfrequency of the limiting interior problem in the case when the corresponding eigenfunction of the limiting problem has a second-order zero at the centre of contraction of the gap. The influence of symmetry of the gap on the absolute value of the imaginary parts of the poles is investigated.

DOI: https://doi.org/10.4213/sm483

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English version:
Sbornik: Mathematics, 2000, 191:6, 821–848

Bibliographic databases:

UDC: 517.956
MSC: Primary 35C20, 35J05; Secondary 35P25, 35B34, 34B27, 78A45
Received: 30.05.1995 and 20.12.1998

Citation: R. R. Gadyl'shin, “On the Dirichlet problem for the Helmholtz equation on the plane with boundary conditions on an almost closed curve”, Mat. Sb., 191:6 (2000), 43–68; Sb. Math., 191:6 (2000), 821–848

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb483
• https://doi.org/10.4213/sm483
• http://mi.mathnet.ru/eng/msb/v191/i6/p43

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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. Konstantin Pankrashkin, “On the spectrum of a waveguide with periodic cracks”, J Phys A Math Theor, 43:47 (2010), 474030
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