Opening a~gap in the essential spectrum of the elasticity problem in a~periodic semi-layer

S. A. Nazarov

Algebra i analiz, Tome 21 (2009) no. 2, pp. 166-204.

Résumé

Rayleigh waves are studied in an elastic half-layer with a periodic end and rigidly clamped faces. It is established that the essential spectrum of the corresponding problem of elasticity theory has a band structure, and an example of a waveguide is presented in which a gap opens in the essential spectrum, i.e., an interval arises that contains points of at most discrete spectrum.

Détail

Keywords: Rayleigh waves, essential spectrum, band structure.

Bibliographie

[1] Rayleigh, Ld., “On waves propagating along the plane surface of an elastic solid”, Proc. London Math. Soc., 17 (1885), 4–11 | DOI

[2] Bonnet-Bendhia A. S., Duterte J., Joly P., “Mathematical analysis of elastic surface waves in topographic waveguides”, Math. Models Methods Appl. Sci., 9:5 (1999), 755–798 | DOI | MR

[3] Kamotskii I. V., Kiselev A. P., “Energeticheskii podkhod k dokazatelstvu suschestvovaniya releevskikh voln v anizotropnom poluprostranstve”, Prikl. mat. i mekh. (to appear)

[4] Nazarov S. A., “Volny Releya dlya uprugogo polusloya s chastichno zaschemlennoi periodicheskoi granitsei”, Dokl. RAN, 423:1 (2008), 56–61

[5] Lekhnitskii S. G., Teoriya uprugosti anizotropnogo tela, Nauka, M., 1977 | MR

[6] Nazarov S. A., Asimptoticheskaya teoriya tonkikh plastin i sterzhnei. Ponizhenie razmernosti i integralnye otsenki, T. 1, Nauch. kn., Novosibirsk, 2002

[7] Ursell F., “Mathematical aspects of trapping modes in the theory of surface waves”, J. Fluid Mech., 183 (1987), 421–437 | DOI | MR | Zbl

[8] Linton C. M., McIver P., “Embedded trapped modes in water waves and acoustics”, Wave Motion, 45 (2007), 16–29 | DOI | MR

[9] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[10] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971 | Zbl

[11] Birman M. Sh., Solomyak M. Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, LGU, L., 1980 | MR

[12] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Tr. Mosk. mat. o-va, 16, 1967, 209–292

[13] Mazya V. G., Plamenevskii B. A., “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach v oblasti s konicheskimi tochkami”, Math. Nachr., 76 (1977), 29–60 | DOI

[14] Mazya V. G., Plamenevskii B. A., “Otsenki v $L_p$ i v klassakh Geldera i printsip maksimuma Miranda–Agmona dlya reshenii ellipticheskikh kraevykh zadach v oblastyakh s osobymi tochkami na granitse”, Math. Nachr., 81 (1978), 25–82 | DOI

[15] Nazarov S. A., “Polinomialnoe svoistvo samosopryazhennykh ellipticheskikh kraevykh zadach i algebraicheskoe opisanie ikh atributov”, Uspekhi mat. nauk, 54:5 (1999), 77–142 | MR | Zbl

[16] Nazarov S. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno gladkoi granitsei\, Nauka, M., 1991

[17] Kozlov V. A., Maz'ya V. G., Rossmann J., Elliptic boundary value problems in domains with point singularities, Math. Surveys Monogr., 52, Amer. Math. Soc., Providence, RI, 1997 | MR | Zbl

[18] Nazarov S. A., “Asimptotika reshenii zadachi teorii uprugosti dlya trekhmernogo tela s tonkimi otrostkami”, Dokl. RAN, 352:4 (1997), 458–461 | MR | Zbl

[19] Nazarov S. A., “Korn's inequalities for junctions of spatial bodies and thin rods”, Math. Methods Appl. Sci., 20:3 (1997), 219–243 | 3.0.CO;2-C class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[20] Kozlov V. A., Maz'ya V. G., Movchan A. B., Asymptotic analysis of fields in multi-structures, Clarendon Press, Oxford Univ. Press, New York, 1999 | MR | Zbl

[21] Nazarov S. A., “Asimptoticheskii analiz i modelirovanie sochleneniya massivnogo tela s tonkimi sterzhnyami”, Tr. Cemin. im. I. G. Petrovskogo, 24, 2004, 95–214

[22] Kuchment P. A., “Teoriya Floke dlya differentsialnykh uravnenii v chastnykh proizvodnykh”, Uspekhi mat. nauk, 37:4 (1982), 3–52 | MR | Zbl

[23] Figotin A., Kuchment P., “Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model”, SIAM J. Appl. Math., 56 (1996), 68–88 | DOI | MR | Zbl

[24] Skriganov M. M., Geometricheskie i arifmeticheskie metody v spektralnoi teorii mnogomernykh periodicheskikh operatorov, Tr. MIAN, 171, M., 1985, 122 pp. | MR | Zbl

[25] Kuchment P., Floquet theory for partial differential equations, Oper. Theory Adv. Appl., 60, Birkhäuser, Basel, 1993 | MR | Zbl

[26] Green E. L., “Spectral theory of Laplace–Beltrami operators with periodic metrics”, J. Differential Equations, 133 (1997), 15–29 | DOI | MR | Zbl

[27] Hempel R., Lineau K., “Spectral properties of periodic media in the large coupling limit”, Comm. Partial Differential Equations, 25 (2000), 1445–1470 | DOI | MR | Zbl

[28] Friedlander L., “On the density of states of periodic media in the large coupling limit”, Comm. Partial Differential Equations, 27 (2002), 355–380 | DOI | MR | Zbl

[29] Zhikov V. V., “O lakunakh v spektre nekotorykh divergentnykh ellipticheskikh operatorov s periodicheskimi koeffitsientami”, Algebra i analiz, 16:5 (2004), 34–58 | MR | Zbl

[30] Filonov N., “Gaps in the spectrum of the Maxwell operator with periodic coefficients”, Comm. Math. Phys., 240:1–2 (2003), 161–170 | DOI | MR | Zbl

[31] Kuchment P., “The mathematics of photonic crystals”, Mathematical Modeling in Optical Science, Frontiers Appl. Math., 22, SIAM, Philadelphia, PA, 2001, 207–272 | MR

[32] Gelfand I. M., “Razlozhenie po sobstvennym funktsiyam uravneniya s periodicheskimi koeffitsientami”, Dokl. AN SSSR, 73:6 (1950), 1117–1120 | MR

[33] Nazarov S. A., “Ellipticheskie kraevye zadachi s periodicheskimi koeffitsientami v tsilindre”, Izv. AN SSSR. Ser. mat., 45:1 (1981), 101–112 | MR | Zbl

[34] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR

[35] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, Uspekhi mat. nauk, 19:3 (1964), 53–161 | MR | Zbl

[36] Nečas J., Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967 | MR | Zbl

[37] Kondratev V. A., Oleinik O. A., “Kraevye zadachi dlya sistemy teorii uprugosti v neogranichennykh oblastyakh. Neravenstva Korna”, Uspekhi mat. nauk, 43:5 (1988), 55–98 | MR | Zbl

[38] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[39] Friedrichs K. O., “On the boundary-value problems of the theory of elasticity and Korn's inequality”, Ann. of Math. (2), 48 (1947), 441–471 | DOI | MR | Zbl

[40] Mosolov P. P., Myasnikov V. P., “Dokazatelstvo neravenstva Korna”, Dokl. AN SSSR, 201:1 (1971), 36–39 | MR | Zbl

[41] Dyuvo G., Lions Zh.-L., Neravenstva v mekhanike i fizike, Nauka, M., 1980 | MR

[42] Shoikhet B. A., “Ob asimptoticheski tochnykh uravneniyakh tonkikh plit slozhnoi struktury”, Prikl. mat. i mekh., 37:5 (1973), 914–924 | MR

[43] Nazarov S. A., “Neravenstva Korna, asimptoticheski tochnye dlya tonkikh oblastei”, Vestn. SPbGU. Ser. 1, 1992, no. 2, 19–24 | MR | Zbl

[44] Nazarov S. A., “Neravenstva Korna dlya uprugikh sochlenenii massivnykh tel, tonkikh plastin i sterzhnei”, Uspekhi mat. nauk, 63:1 (2008), 37–110 | MR | Zbl

[45] Nazarov S. A., “Obosnovanie asimptoticheskoi teorii tonkikh sterzhnei. Integralnye i potochechnye otsenki”, Probl. mat. anal., 17, SPbGU, SPb., 1997, 101–152

[46] Nazarov S. A., “Neravenstvo Korna dlya uprugogo soedineniya tela so sterzhnem”, Problemy mekhaniki deformiruemogo tverdogo tela, SPbGU, SPb., 2002, 234–240

[47] Nazarov S. A., “Lovushechnye mody dlya tsilindricheskogo uprugogo volnovoda s dempfiruyuschei prokladkoi”, Zh. vychisl. mat. i mat. fiz., 48:5 (2008), 863–881 | MR | Zbl