Abstract
General formally self-adjoint boundary value problems with spec tral parameter are investigated in domains with cylindrical and quasicylin drical (periodic) outlets to infinity. The structure of the spectra is studied for operators generated by the corresponding sesquilinear forms. In addition to general results, approaches and methods are discussed to get a piece of infor mation on continuous, point, and discrete spectra, in particular, for specific problems in the mathematical physics.
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References
Birman, M.S., Solomyak, M.Z.: Spectral Theory of Self-Adjoint operators in Hilbert Space. Reidel Publ. Company, Dordrecht (1986)
Bonnet-Bendhia, A.S., Duterte, J., Joly, P.: Mathematical analysis of elastic surface waves in topographic waveguides. Math. Meth. Appl. Sci.9, no. 5, 755–798 (1999)
Bonnet-Bendhia, A.S., Starling, F.: Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Meth. Appl. Sci.77, 305–338 (1994)
Cherepanov, G.P.: Crack propagation in continuous media (Russian). Prikl. Mat. Mekh.31, 476–488 (1967); English transl.: J. Appl. Math. Mech.31, 503–512 (1967)
Costabel, M., Dauge, M.: Crack singularities for general elliptic systems. Math. Nachr.235, 29–49 (2002)
Duduchava, R., Wendland, W.L.: The Wiener-Hopf method for systems of pseudod ifferential equations with an application to crack problems. Integral Eq. Operator Theory23, no. 3, 294–335 (1995)
Duvaut, G., Lions. J.L.: Les inéquations en mćanique et en physique. Dunod, Paris (1972)
Evans, D.V., Levitin, M., Vasil'ev, D.: Existence theorems for trapped modes. J. Fluid Mech.261, 21–31 (1994)
Figotin, A., Kuchment, P.: Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model. SIAM J. Appl. Math.56, 68–88 (1996); II. Two-dimensional photonic crystals. ibid.56, 1561–1620 (1996)
Filonov, N.: Gaps in the spectrum of the Maxwell operator with periodic coefficients. Commun. Math. Phys.240, no. 1–2, 161–170 (2003)
Friedlander, L.: On the density of states of periodic media in the large coupling limit. Commun. Partial Differ. Equ.27, 355–380 (2002)
Gel'fand, I.M.: Expansion in characteristic functions of an equation with periodic coefficients (Russian). Dokl. Akad. Nauk SSSR73, 1117–1120 (1950)
Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Am. Math. Soc., Providence, RI (1969)
Gohberg, I.C., Sigal, E.I.: An operator generalization of the logarithmic residue the orem and the theorem of Rouché (Russian). Mat. Sb.84, 607–629. (1971); English transl.: Math. USSR-Sb.13, 603–625 (1971)
Green, E.L.: Spectral theory of Laplace-Beltrami operators with periodic metrics. J. Differ. Equations133, 15–29 (1997)
Grinchenko, V.T., Ulitko, A.F., Shul'ga, N.A.: Mechanics of Coupled Fields in Struc tural Elements (Russian). Naukova Dumka, Kiev (1989)
Hempel, R., Lineau, K.: Spectral properties of the periodic media in large coupling limit. Commun. Partial Differ. Equ.25, 1445–1470 (2000)
Jones, D.S.: The eigenvalues of ▿2 u+ λu= 0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc.49, 668–684 (1953)
Kamotskii, I.V., Nazarov, S.A.: Elastic waves localized near periodic sets of flaws (Russian). Dokl. Ross. Akad. Nauk.368, no. 6, 771–773 (1999); English transl.: Dokl. Physics44, no. 10, 715–717 (1999)
Kamotskii, I.V., Nazarov, S.A.: On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain (Russian). Probl. Mat. Anal.19, 105–148 (1999); English transl.: J. Math. Sci.101, no. 2, 2941–2974 (2000)
Kamotskii, I.V., Nazarov, S.A.: Exponentially decreasing solutions of diffraction prob lems on a rigid periodic boundary (Russian). Mat. Zametki73, no. 1, 138–140 (2003); English transl.: Math. Notes.73, no. 1, 129–131 (2003)
Kato, T.: Perturbation Theory for Linear Operators. Springer (1966)
Khallaf, N.S.A., Parnovski, L., Vassiliev, D.: Trapped modes in a waveguide with a long obstacle. J. Fluid Mech.403, 251–261 (2000)
Knowles, J.K., Sternberg, E.: On a class of conservation laws in linearizad and finite elastostatics. Arch. Ration. Mech. Anal.44, no. 3, 187–211 (1972)
Kondratiev, V.A.: Boundary problems for elliptic equations in domains with conical or angular points (Russian). Tr. Mosk. Mat. Obshch.16, 209–292 (1967); English transl.: Trans. Mosc. Math. Soc.16, 227–313 (1967)
Kondratiev, V.A., Oleinik, O.A.: Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities (Russian). Uspekhi Mat. Nauk43, no. 5, 55–98 (1988); English transl.: Russian Math. Surveys43, no. 5, 65–119 (1988)
Korn, A.: Solution générale du probléme déquilibre dans la théorie lélasticité dans le cas ou` les efforts sont donnés à la surface. Ann. Univ. Toulouse, 165–269 (1908)
Kozlov, V.A., Maz'ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. Am. Math. Soc., Providence, RI (1997)
Kuchment, P.A.: Floquet theory for partial differential equations (Russian). Uspekhi Mat. Nauk37, no. 4, 3–52 (1982); English transl.: Russian Math. Surveys37, no. 4, 1–60 (1982)
Kuchment, P.: Floquet Theory for Partial Differential Equations. Birch¨auser, Basel (1993)
Kuchment, P.: The mathematics of photonic crystals. In: Mathematical Modeling in Optical Science, pp. 207–272. SIAM (2001)
Kulikov, A.A., Nazarov, S.A.: Cracks in piezoelectric and electro-conductive bodies (Russian). Sib. Zh. Ind. Mat.8, no. 1, 70–87 (2005)
Ladyzhenskaya, O.A.: Boundary Value Problems of Mathematical Physics. Springer-Verlag, New-York (1985)
Lekhnitskij, S.G.: Theory of Elasticity of an Anisotropic Body (Russian). Nauka, Moscow (1977); English transl.: Mir Publishers, Moscow, 1981.
Lions. J.L., Magenes. E.:Non-Homogeneous Boundary Value Problems and Ap plications(French). Dunod, Paris (1968); English transl.: Springer-Verlag, Berlin-Heidelberg-New York (1972)
Maz'ja, V.G., Plamenevskii, B.A.: On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr.76, 29–60 (1977); English transl.: Am. Math. Soc. Transl. (Ser. 2)123, 57–89 (1984)
Maz'ja, V.G., Plamenevskii, B.A.: Estimates inLpand Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value prob lems in domains with singular points on the boundary. Math. Nachr.81, 25–82 (1978); English transl.: Am. Math. Soc. Transl. (Ser. 2)123, 1–56 (1984)
Nazarov, S.A.: Elliptic boundary value problems with periodic coefficients in a cylin der (Russian). Izv. Akad. Nauk SSSR. Ser. Mat.45, no. 1, 101–112 (1981); English transl.: Math. USSR.-Izv.18, no. 1, 89–98 (1982)
Nazarov, S.A.: A general scheme for averaging self-adjoint elliptic systems in multi dimensional domains, including thin domains (Russian). Algebra Anal.7, no. 5, 1–92 (1995); English transl.: St. Petersbg. Math. J.7, no. 5, 681–748 (1996)
Nazarov, S.A.: Korn's inequalities for junctions of spatial bodies and thin rods. Math. Methods Appl. Sci.20, no. 3, 219–243 (1997)
Nazarov, S.A.: Self-adjoint elliptic boundary-value problems. The polynomial prop erty and formally positive operators (Russian). Probl. Mat. Anal.16, 167–192 (1997); English transl.: J. Math. Sci.92, no. 6, 4338–4353 (1998)
Nazarov, S.A.: Weight functions and invariant integrals (Russian). Vychisl. Mekh. Deform. Tverd. Tela.1, 17–31 (1990)
Nazarov, S.A.: The interface crack in anisotropic bodies. Stress singularities and invariant integrals (Russian). Prikl. Mat. Mekh.62, no. 3, 489–502 (1998); English transl.: J. Appl. Math. Mech.62, no. 3, 453–464 (1998)
Nazarov, S.A.: The polynomial property of self-adjoint elliptic boundary-value prob lems and the algebraic description of their attributes (Russian). Uspekhi Mat. Nauk54, no. 5, 77–142 (1999); English transl.: Russian Math. Surveys54, no. 5, 947–1014 (1999)
Nazarov, S.A.: Asymptotic Theory of Thin Plates and Rods. Vol.1. Dimension Reduc tion and Integral Estimates (Russian). Nauchnaya Kniga IDMI, Novosibirsk (2002)
Nazarov, S.A.: Korn's inequality for an elastic junction of a body with a rod (Russian). In: Problems of Mechanics of Solids, pp. 234–240. St.-Petersburg State Univ. Press, St.-Petersburg (2002)
Nazarov, S.A.: Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate (Russian). Probl. Mat. Anal.25, 99–188 (2003); English transl.: J. Math. Sci.114, no. 5, 1657–1725 (2003)
Nazarov, S.A.: Trapped modes in a cylindrical elastic waveguide with a damping gasket (Russian). Zh. Vychisl. Mat. Mat. Fiz.48, no. 5, 863–881 (2008); English transl.: Comput. Math. Math. Phys.48, no. 5, 816–833 (2008)
Nazarov, S.A.: Korn inequalities for elastic junctions of massive bodies, thin plates and rods. inequalities (Russian). Uspekhi Mat. Nauk63, no. 5, 37–110 (2008); English transl.: Russian Math. Surveys63, no. 5 (2008)
Nazarov, S.A.: The Rayleigh waves in an elastic semi-layer with periodic boundary. Dokl. Ross. Akad. Nauk [Submitted]
Nazarov, S.A.: On the essential spectrum of boundary value problems for systems of differential equations in a bounded domain with a peak. Funkt. Anal. Pril. [To appear]
Nazarov, S.A.: Opening gaps in the continuous spectrum of periodic elastic waveguide with traction-free boundary. Zh. Vychisl. Mat. Mat. Fiz. [Submitted]
Nazarov, S.A., Plamenevskii, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin, New York (1994)
Nečas. J.: Les méthodes in théorie des equations elliptiques. Masson-Academia, Paris-Prague (1967)
Parton, V.Z., Kudryavtsev, B.A.: Electromagnetoelasticity of Piezoelectronics and Conductive Solids (Russian). Nauka, Moscow (1988)
Rellich, F.: Über das asymptotische Verhalten der Lösungen von ▿2 u+ λu= 0 in unendlichen Gebieten. J. Dtsch. Math.-Ver.53, no. 1, 57–65 (1943)
Rice, J.R.: A path independent integral and the approximate analysis of strain con centration by notches and cracks. J. Appl. Mech.35, no. 2, 379–386 (1968)
Roitberg, I., Vassiliev, D., Weidl, T.: Edge resonance in an elastic semi-strip. Quart. J. Appl. Math.51, no. 1, 1–13 (1998)
Skriganov, M.M.: Geometric and arithmetical methods in the spectral theory of mul tidimensional periodic operators (Russian). Tr. Mat. Inst. Steklova171(1985)
Sobolev, S.L.: Some Applications of Functional Analysis in Mathematical Physics (Russian). 3rd Ed. Nauka, Moscow (1988); English transl.: Am. Math. Soc, Provi dence, RI (1991)
Sukhinin, S.V.: Waveguide, anomalous, and whispering properties of a periodic chain of obstacles (Russian). Sib. Zh. Ind. Mat.1, no. 2, 175–198 (1998)
Zhikov, V.V.: Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients (Russian). Algebra Anal.16, no. 5, 34–58 (2004); English transl.: St. Petersbg. Math. J.16, no. 5, 773–790 (2005)
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Nazarov, S. (2009). Properties of Spectra of Boundary Value Problems in Cylindrical and Quasicylindrical Domains. In: Maz'ya, V. (eds) Sobolev Spaces in Mathematics II. International Mathematical Series, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85650-6_12
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