Abstract:
We consider the Neumann boundary-value problem of finding the
small-parameter asymptotics of the eigenvalues and eigenfunctions for the
Laplace operator in a singularly perturbed domain consisting of two
bounded domains joined by a thin “handle”. The small parameter is the
diameter of the cross-section of the handle. We show that as the small
parameter tends to zero these eigenvalues converge either to the
eigenvalues corresponding to the domains joined or to the eigenvalues of
the Dirichlet problem for the Sturm–Liouville operator on the segment to
which the thin handle contracts. The main results of this paper
are the complete power small-parameter asymptotics of the eigenvalues and the
corresponding eigenfunctions and explicit formulae for the first terms of
the asymptotics. We consider critical cases generated by the choice of the
place where the thin “handle” is joined to the domains, as well as by
the multiplicity of the eigenvalues corresponding to the domains joined.
\Bibitem{Gad05}
\by R.~R.~Gadyl'shin
\paper On the eigenvalues of a ``dumb-bell with a thin handle''
\jour Izv. Math.
\yr 2005
\vol 69
\issue 2
\pages 265--329
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This publication is cited in the following 25 articles:
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S. A. Nazarov, “Abnormal Transmission of Elastic Waves through a Thin Ligament Connecting Two Planar Isotropic Waveguides”, Mech. Solids, 57:8 (2022), 1908
Lucas Chesnel, Jérémy Heleine, Sergei A. Nazarov, “Acoustic passive cloaking using thin outer resonators”, Z. Angew. Math. Phys., 73:3 (2022)
Bucur D., Henrot A., Michetti M., “Asymptotic Behaviour of the Steklov Spectrum on Dumbbell Domains”, Commun. Partial Differ. Equ., 46:2 (2021), 362–393
Chesnel L., Nazarov S.A., “Design of An Acoustic Energy Distributor Using Thin Resonant Slits”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 477:2247 (2021), 20200896
S. A. Nazarov, L. Chesnel, “Anomalies of acoustic wave propagation in two semi-infinite cylinders connected by a flattened ligament”, Comput. Math. Math. Phys., 61:4 (2021), 646–663
S. A. Nazarov, “Waveguide with double threshold resonance at a simple threshold”, Sb. Math., 211:8 (2020), 1080–1126
Chesnel L., Nazarov S.A., Taskinen J., “Surface Waves in a Channel With Thin Tunnels and Wells At the Bottom: Non-Reflecting Underwater Topography”, Asymptotic Anal., 118:1-2 (2020), 81–122
Nazarov S.A., “Anomalies of Acoustic Wave Scattering Near the Cut-Off Points of Continuous Spectrum (a Review)”, Acoust. Phys., 66:5 (2020), 477–494
A. L. Delitsyn, “Localization of eigenfunctions of the Laplace operator in a domain with a perforated barrier”, Comput. Math. Math. Phys., 59:6 (2019), 936–941
Bonnet-Ben Dhia A.-S., Chesnel L., Nazarov S.A., “Perfect Transmission Invisibility For Waveguides With Sound Hard Walls”, J. Math. Pures Appl., 111 (2018), 79–105
Delitsyn A., Grebenkov D.S., “Mode Matching Methods For Spectral and Scattering Problems”, Q. J. Mech. Appl. Math., 71:4 (2018), 537–580
Arrieta J.M., Ferraresso F., Lamberti P.D., “Spectral Analysis of the Biharmonic Operator Subject to Neumann Boundary Conditions on Dumbbell Domains”, Integr. Equ. Oper. Theory, 89:3 (2017), 377–408
F. L. Bakharev, S. A. Nazarov, “Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions”, Siberian Math. J., 56:4 (2015), 575–592
Bunoiu R., Cardone G., Nazarov S.A., “Scalar Boundary Value Problems on Junctions of Thin Rods and Plates”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 48:5 (2014), 1495–1528
S. A. Nazarov, “Asymptotics of eigen-oscillations of a massive elastic body with a thin baffle”, Izv. Math., 77:1 (2013), 87–142
D. S. Grebenkov, B.-T. Nguyen, “Geometrical Structure of Laplacian Eigenfunctions”, SIAM Rev, 55:4 (2013), 601
Wang Rong-Nian, Chen De-Han, Xiao Ti-Jun, “Abstract fractional Cauchy problems with almost sectorial operators”, J. Differential Equations, 252:1 (2012), 202–235
S. A. Nazarov, “Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler”, Siberian Math. J., 53:2 (2012), 274–290
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