Umov-Mandelshtam radiation conditions in elastic periodic waveguides

S. A. Nazarov

Sbornik. Mathematics, Tome 205 (2014) no. 7, pp. 953-982

Résumé

We study settings of the problem of elasticity theory on wave propagation in an elastic periodic waveguide with radiation conditions at infinity. We present a mathematical theory for energy radiation conditions based on Mandelshtam's energy principle and the Umov-Poynting vector, as well as using the technique of weighted spaces with detached asymptotics and the energy transfer symplectic form. We establish that in a threshold situation, that is, when standing and polynomial elastic Floquet waves appear, the well-known limiting absorption principle, in contrast to the energy principle that is being applied, cannot identify the direction of the wave's motion. Bibliography: 37 titles.

Détail

Keywords: elastic periodic waveguide, Mandelshtam's energy radiation condition, Umov-Poynting vector, energy transfer symplectic form.

Bibliographie

[1] S. G. Lekhnitskii, Theory of elasticity of an anisotropic body, Mir, M., 1981, 431 pp. | MR | MR | Zbl | Zbl

[2] S. A. Nazarov, Asimptoticheskaya teoriya tonkikh plastin i sterzhnei, v. 1, Ponizhenie razmernosti i integralnye otsenki, Nauchn. kniga, Novosibirsk, 2002, 406 pp.

[3] V. A. Kondrat'ev, O. A. Oleinik, “Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities”, Russian Math. Surveys, 43:5 (1988), 65–119 | DOI | MR | Zbl

[4] S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107 | DOI | DOI | MR | Zbl

[5] I. I. Vorovich, V. A. Babeshko, Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblastei, Nauka, M., 1979, 320 pp. | MR | Zbl

[6] V. A. Babeshko, E. V. Glushkov, Zh. F. Zinchenko, Dinamika neodnorodnykh lineinouprugikh sred, Nauka, M., 1989, 343 pp. | Zbl

[7] C. A. Nazarov, “Energeticheskie usloviya izlucheniya Mandelshtama i vektor Umova–Pointinga v uprugikh volnovodakh”, Problemy matem. analiza, 2013, no. 72, 101–146

[8] S. A. Nazarov, B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Exp. Math., 13, Walter de Gruyter Co., Berlin, 1994, viii+525 pp. | DOI | MR | Zbl

[9] N. A. Umov, Uravneniya dvizheniya energii v telakh, Tip. Ulrikha i Shultse, Odessa, 1874, 58 pp.

[10] V. G. Maz'ya, B. A. Plamenevskij, “Estimates in $L_p$ and in Hölder classes and the Miranda–Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary”, Amer. Math. Soc. Transl. Ser. 2, 123, Amer. Math. Soc., Providence, RI, 1984, 1–56 | DOI | MR | Zbl

[11] S. A. Nazarov, “Elliptic boundary value problems with periodic coefficients in a cylinder”, Izv. Akad. Nauk SSSR Ser. Mat., 18:1 (1982), 89–98 | DOI | MR | Zbl

[12] P. A. Kuchment, “Floquet theory for partial differential equations”, Russian Math. Surveys, 37:4 (1982), 1–60 | DOI | MR | Zbl

[13] S. A. Nazarov, “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains”, Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), 9, Springer, New York, 2009, 261–309 | DOI | MR | Zbl

[14] M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators”, Proc. Steklov Inst. Math., 171 (1987), 1–121 | MR | Zbl

[15] P. Kuchment, Floquet theory for partial differential equations, Oper. Theory Adv. Appl., 60, Birchäuser, Basel, 1993, xiv+350 pp. | DOI | MR | Zbl

[16] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Tr. MMO, 16, Izd-vo Mosk. un-ta, M., 1967, 209–292 | MR | Zbl

[17] I. M. Gelfand, “Razlozhenie po sobstvennym funktsiyam uravneniya s periodicheskimi koeffitsientami”, Dokl. AN SSSR, 73 (1950), 1117–1120 | MR | Zbl

[18] E. Hille, R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., 31, rev. ed., Amer. Math. Soc., Providence, R.I., 1957, xii+808 pp. | MR | MR | Zbl

[19] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag, New York, 1966, xix+592 pp. | MR | MR | Zbl | Zbl

[20] I. C. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, R.I., 1969, xv+378 pp. | MR | MR | Zbl | Zbl

[21] M. Sh. Birman, M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, Math. Appl. (Soviet Ser.), D. Reidel Publishing Co., Dordrecht, 1987, xv+301 pp. | MR | MR | Zbl

[22] S. A. Nazarov, B. A. Plamenevskii, “On radiation conditions for selfadjoint elliptic problems”, Soviet Math. Dokl., 41:2 (1991), 274–277 | MR | Zbl

[23] C. A. Nazarov, B. A. Plamenevskii, “Printsipy izlucheniya dlya samosopryazhennykh ellipticheskikh zadach”, Differentsialnye uravneniya. Spektralnaya teoriya. Rasprostranenie voln, Problemy matem. fiziki, 13, Izd-vo Leningr. un-ta, L., 1991, 192–244 | MR | Zbl

[24] S. A. Nazarov, “Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions”, Amer. Math. Soc. Transl. Ser. 2, 193, Amer. Math. Soc., Providence, RI, 1999, 77–125 | MR | Zbl

[25] J. H. Poynting, “On the transfer of energy in the electromagnetic field”, Phil. Trans. R. Soc. Lond., 175 (1884), 343–361 | DOI | Zbl

[26] L. I. Mandelshtam, Lektsii po optike, teorii otnositelnosti i kvantovoi mekhanike, Polnoe sobranie trudov, 5, Izd-vo AN SSSR, M., 1950, 438 pp. | MR

[27] S. Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965, xvii+508 pp. | MR | Zbl | Zbl

[28] A. T. Fomenko, Symplectic geometry, Adv. Stud. Contemp. Math., 5, Gordon and Breach Science Publishers, New York, 1988, xiv+387 pp. | MR | MR | Zbl | Zbl

[29] C. A. Nazarov, “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach s periodicheskimi koeffitsientami”, Vestn. LGU. Ser. 1, 1985, no. 3 (No 15), 16–22 | MR | Zbl

[30] V. G. Maz'ya, B. A. Plamenevskij, “On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points”, Amer. Math. Soc. Transl. Ser. 2, 123, Amer. Math. Soc., Providence, RI, 1984, 57–88 | DOI | MR | Zbl

[31] S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide”, Theoret. and Math. Phys., 167:2 (2011), 606–627 | DOI | DOI | Zbl

[32] I. V. Kamotskii, S. A. Nazarov, “The augmented scattering matrix and exponentially decaying solutions of an elliptic problem in a cylindrical domain”, J. Math. Sci. (New York), 111:4 (2002), 3657–3666 | DOI | MR | Zbl

[33] M. M. Vainberg, V. A. Trenogin, Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969, 567 pp. | MR | Zbl

[34] A. G. Sveshnikov, “O printsipe izlucheniya”, Dokl. AN SSSR, 73:5 (1950), 917–920 | MR | Zbl

[35] A. G. Sveshnikov, “Printsip predelnogo pogloscheniya dlya volnovoda”, Dokl. AN SSSR, 80:3 (1951), 345–347 | MR | Zbl

[36] D. M. Eidus, “Printsip predelnogo pogloscheniya v teorii uprugosti”, Vestn. LGU, ser. matem. mekh., 1963, no. 7, 141–154

[37] R. D. Mindlin, “Waves and vibration in isotropic elastic plates”, 1960 Structural mechanics, Pergamon Press, New York, 1960, 199–232 | MR