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Keywords:
spectral problem; thin domain; boundary layer; trapped mode; localized eigenfunction
Summary:
It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain $\Omega _h$ is localized either at the whole lateral surface $\Gamma _h$ of the domain, or at a point of $\Gamma _h$, while the eigenfunction decays exponentially inside $\Omega _h$. Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.
References:
[1] Ciarlet P. G., Kesavan S.: Two dimensional approximations of three dimensional eigenvalues in plate theory. Comput. Methods Appl. Mech. Engrg. 26 (1980), 149–172. MR 0626720
[2] Zorin I. S., Nazarov S. A.: Edge effect in the bending of a thin three-dimensional plate. J. Appl. Math. Mech. 53 (1989), 500–507. MR 1022416
[3] Dauge M., Djurdjevic I., Faou E., Rössle A.: Eigenmode asymptotics in thin elastic plates. J. Math. Pures Appl. 78 (1999), 925–964. MR 1725748
[4] Berdichevskii V. L.: High-frequency long-wave oscillations of plates. Doklady AN SSSR 236 (1977), 1319–1322. MR 0455709
[5] Berdichevskii V. L.: Variational Principles in Mechanics of Continuous Media. Nauka, Moskva, 1983. MR 0734171
[6] Nazarov S. A.: On the asymptotics of the spectrum of a thin plate problem of elasticity. Siberian Math. J. 41 (2000), 744–759. MR 1785611 | Zbl 1150.74367
[7] Nazarov S. A.: Asymptotics of eigenvalues of the Dirichlet problem in a thin domain. Sov. Math. 31 (1987), 68–80. Zbl 0664.35064
[8] Kamotskii I. V., Nazarov S. A.: On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain. Probl. matem. analiz 19 (1999), 105–148. (Russian) MR 1784687
[9] Maz’ya V., Nazarov S., Plamenevskij B.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. 1, 2. Birkhäuser, Basel, 2000.
[10] Evans D. V., Levitin M., Vasil’ev D.: Existence theorems for trapped modes. J. Fluid Mech. 261 (1994), 21–31. MR 1265871
[11] Roitberg I., Vassiliev D., Weidl T.: Edge resonance in an elastic semi-strip. Q. J. Mech. Appl. Math. 51 (1998), 1–13. MR 1610688
[12] Nazarov S. A.: The structure of solutions of elliptic boundary value problems in slender domains. Vestn. Leningr. Univ. Math. 15 (1983), 99–104. Zbl 0527.35011
[13] Nazarov S. A.: A general scheme for averaging selfadjoint elliptic systems in multidimensional domains, including thin domains. St. Petersburg Math. J. 7 (1996), 681–748. MR 1365812
[14] Nazarov S. A.: Singularities of the gradient of the solution of the Neumann problem at the vertex of a cone. Math. Notes 42 (1987), 555–563. MR 0910031 | Zbl 0639.35018
[15] Maz’ya V. G., Nazarov S. A., Plamenevskii B. A.: On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone. Math. USSR Sbornik 50 (1985), 415–437.
[16] Nazarov S. A.: Justification of asymptotic expansions of the eigenvalues of non-selfadjoint singularly perturbed elliptic boundary value problems. Math. USSR Sbornik 57 (1987), 317–349. MR 0837128
[17] Nazarov S. A.: Asymptotic Theory of Thin Plates and Rods. Dimension Reduction and Integral Estimates. Nauchnaya Kniga, Novosibirsk, 2001. (Russian)
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