Abstract
For the linear theory of water waves, we find out families of submerged or surface-piercing bodies in an infinite three-dimensional canal, which depend on a small parameter ε > 0 and have the following property: for any positive d and natural J, there exists ε(d, J) > 0 such that, for ε ∈ (0, ε(d, J)], the segment [0, d] of the continuous spectrum of the problem contains at least J eigenvalues. These eigenvalues are associated with trapped modes, i.e., solutions of the homogeneous problem, which decay exponentially at infinity and possess finite energy. Bibliography: 27 titles.
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References
N. Kuznetsov, V. Maz’ya, and B. Vainberg, Linear Water Waves, Cambridge University Press, Cambridge (2002).
O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics [in Russian], Moscow (1973).
D. V. Evans, M. Levitin, and D. Vasil’ev, “Existence theorems for trapped modes,” J. Fluid Mech., 261, 21–31 (1994).
I. Roitberg, D. Vassiliev, and T. Weidl, “Edge resonance in an elastic semi-strip,” Quart. J. Appl. Math., 51, 1–13 (1998).
S. A. Nazarov, “Trapped modes for a cylindrical elastic waveguide with a damping gasket,” Zh. Vychisl. Mat. Mat. Fiz., 48 (2008).
C. M. Linton and P. McIver, “Embedded trapped modes in water waves and acoustics,” Wave Motion, 45, 16–29 (2007)
I. V. Kamotskii and S. A. Nazarov, “Wood’s anomalies and surface waves in the problem of scattering by a periodic boundary. 1,” Mat. Sb., 190, No. 1, 109–138; No. 2, 43–70 (1999).
I. V. Kamotskii and S. A. Nazarov, “The augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain,” Zap. Nauchn. Semin. POMI, 264, 66–82 (2000).
S. A. Nazarov, “A criterion for the existence of decaying solutions in the problem on a resonator with a cylindrical waveguide,” Funkt. Anal. Prilozh., 40, 30–32 (2006).
S. A. Nazarov, “Artificial boundary conditions for finding surface waves in the problem of diffraction by a periodic boundary,” Zh. Vychisl. Mat. Mat. Fiz., 46, 2267–2278 (2006).
S. N. Leora, S. A. Nazarov, and A. V. Proskura, “Derivation of limiting equations for elliptic problems in thin domains using computers,” Zh. Vychisl. Mat. Mat. Fiz., 26, 1032–1048 (1986).
S. A. Nazarov, “A general scheme for averaging self-adjoint elliptic systems in multidimensional domains, including thin domains,” Algebra Analiz, 7, 1–92 (1995).
S. A. Nazarov, Asymptotic Theory of Thin Plates and Rods. Vol. 1. Dimension Reduction and Integral Estimates [in Russian], Novosibirsk (2002).
M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Reidel Publ. Company, Dordrecht (1986).
M. I. Vishik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter,” Usp. Mat. Nauk, 12, 3–122 (1957).
S. A. Nazarov, “Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate,” Probl. Mat. Analiz, 25, 99–188 (2003).
M. Lobo, S. A. Nazarov, and E. Perez, “Eigen-oscillations of contrasting nonhomogeneous bodies: asymptotic and uniform estimates for eigenvalues,” IMA J. Appl. Math., 70, 419–458 (2005).
D. Gomez, M. Lobo, S. A. Nazarov, and E. Perez, “Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues,” J. Math. Pures Appl., 85, 598–632 (2006).
O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Amsterdam, North-Holland (1992).
V. A. Kondratiev, “Boundary problems for elliptic equations in domains with conic or angular points,” Trudy Moskov. Mat. Obshch., 16, 209–293 (1967).
V. G. Maz’ja and B. A. Plamenevskii, “On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points,” Math. Nach., 76, 29–60 (1977).
V. G. Maz’ja and B. A. Plamenevskii, “Estimates in L p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary,” Math. Nachr., 81, 25–82 (1978).
S. A. Nazarov, “The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes,” Usp. Mat. Nauk, 54, 77–142 (1999).
S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin, New York (1994).
S. A. Nazarov, “Self-adjoint extensions of the Dirichlet problem operator in weighted function spaces,” Mat. Sb., 137, 224–241 (1988).
S. A. Nazarov, “Asymptotic conditions at a point, self-adjoint extensions of operators and the method of matched asymptotic expansions,” Trudy Peterb. Mat. Obshch., 5, 112–183 (1996).
S. A. Nazarov, “Asymptotic behavior with respect to a small parameter of the solution to a boundary value problem that is elliptic in the sense of Agranovich-Vishik in a domain with a conic point,” Probl. Mat. Anal., 7, 146–167 (1979).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 98–126.
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Nazarov, S.A. Concentration of the point spectrum on the continuous one in problems of linear water-wave theory. J Math Sci 152, 674–689 (2008). https://doi.org/10.1007/s10958-008-9095-2
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DOI: https://doi.org/10.1007/s10958-008-9095-2