## Bands in the spectrum of a periodic elastic waveguide.(English)Zbl 1378.35295

Summary: We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order $$h >0$$. The essential spectrum of the problem is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as $$h \rightarrow 0$$.

### MSC:

 35Q74 PDEs in connection with mechanics of deformable solids 35J57 Boundary value problems for second-order elliptic systems 74J05 Linear waves in solid mechanics
Full Text:

### References:

 [1] Bakharev, FL; Nazarov, SA, Gaps in the spectrum of a waveguide composed of domains with different limiting dimensions, Sib. Math. J., 56, 575-592, (2015) · Zbl 1327.35262 [2] Bakharev, FL; Nazarov, SA; Ruotsalainen, K, On the spectrum of Neumann-Laplacian on a cylinder with periodically immersed obstacles, Appl. Anal., 92, 1889-1915, (2013) · Zbl 1302.35266 [3] Bakharev, FL; Ruotsalainen, K; Taskinen, J, Spectral gaps for the linear surface wave model in periodic channels, Q. J. Mech. Appl. Math., 67, 343-362, (2014) · Zbl 1302.76028 [4] Birman, M.S., Solomyak, M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. Reidel Publishing Company, Dordrecht (1986) · Zbl 0744.47017 [5] Cardone, G; Minutolo, V; Nazarov, SA, Gaps in the essential spectrum of periodic elastic waveguides, Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 89, 729-741, (2009) · Zbl 1189.35201 [6] Cardone, G; Nazarov, SA; Perugia, C, A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface, Math. Nachr., 283, 1222-1244, (2010) · Zbl 1213.35327 [7] 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) · Zbl 0852.35014 [8] Figotin, A; Kuchment, P, Band-gap structure of spectra of periodic dielectric and acoustic media. II. two-dimensional photonic crystals, SIAM J. Appl. Math., 56, 1561-1620, (1996) · Zbl 0868.35009 [9] Filonov, N, Gaps in the spectrum of the Maxwell operator with periodic coefficients, Commun. Math. Phys., 240, 161-170, (2003) · Zbl 1037.35051 [10] Friedlander, L, On the density of states of periodic media in the large coupling limit, Commun. Partial Differ. Equ., 27, 355-380, (2002) · Zbl 1055.35083 [11] Friedlander, L; Solomyak, M, On the spectrum of narrow periodic waveguides, Russ. J. Math. Phys., 15, 238-242, (2008) · Zbl 1180.35392 [12] Gelfand, IM, Expansion in characteristic functions of an equation with periodic coefficients, Dokl. Akad. Nauk. SSSR, 73, 1117-1120, (1950) [13] Green, EL, Spectral theory of Laplace-Beltrami operators with periodic metrics, J. Differ. Equ., 133, 15-29, (1997) · Zbl 0924.35087 [14] Hempel, R; Lineau, K, Spectral properties of the periodic media in large coupling limit, Commun. Partial Differ. Equ., 25, 1445-1470, (2000) · Zbl 0958.35089 [15] Il’in, A.M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Translations of Mathematical Monographs, vol. 102. American Mathematical Society, Providence (1992) [16] Kondratiev, VA, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Mosk. Mat. Obsc., 16, 209-292, (1967) [17] Kondratiev, V.A., Oleinik, O.A.: On Korn’s inequalities. C. R. Acad. Sci. Paris Ser. I 308, 483-487 (1989) · Zbl 0698.35067 [18] Kuchment, P.: Floquet theory for partial differential equations. Uspekhi Mat. Nauk. 37(4), 3-52 (1982). (in Russian); (English transl: Russ. Math. Surv. 37(4), 1-60 (1982)) [19] Ladyshenskaja, O.A.: Boundary Value Problems of Mathematical Physics. Springer, New York (1985) [20] Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 7. Theory of Elasticity. Translated from the Russian by J. B. Sykes and W. H. Reid. 3rd edn. Pergamon Press, Oxford (1986) · Zbl 0043.19803 [21] Lekhnitskii, S.G.: Elasticity of an Anisotropic Body. Nauka, Moscow (1977). (in Russian) · Zbl 0467.73012 [22] Maz’ja, VG; Plamenevskii, BA, The coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points, Math. Nachr., 76, 29, (1977) · Zbl 0359.35024 [23] Nazarov, S.A.: Elliptic boundary value problems with periodic coefficients in a cylinder. Izv. Akad. Nauk SSSR. Ser. Mat. 45(1), 101-112 (1981) (English transl. Math. USSR. Izvestija. 18(1), 89-98 (1982)) · Zbl 0462.35034 [24] Nazarov, S.A.: Polynomial property of self-adjoint elliptic boundary value problems, and the algebraic description of their attributes. Uspekhi Mat. Nauk. 54(5(329)), 77-142 (1999). (in Russian); translation Russ. Math. Surv. 54(5), 9414 (1999) · Zbl 1158.74407 [25] Nazarov, S.A.: Properties of Spectra of Boundary Value Problems in Cylindrical and Quasicylindrical Domains, Sobolev Spaces in Mathematics, vol. 2 (Maz’ya, V. ed.). International Mathematical Series 9, pp. 261-309 (2008) · Zbl 1208.35093 [26] Nazarov, S.A.: Asymptotics of solutions to the spectral Steklov problem in a domain with a blunted peak. Mat. Zametki. 86, 4 (2009). (in Russian) (English transl.: Math. Notes. 84, 3-4 (2009)) · Zbl 1213.35327 [27] Nazarov, SA, A gap in the continuous spectrum of an elastic waveguide, C. R. Mec., 336, 751-756, (2008) · Zbl 1158.74407 [28] Nazarov, S.A.L: The Rayleigh waves in an elastic half-layer with partly jammed periodic boundary. Dokl. Ross. Akad. Nauk. 423(1), 56-61 (2008). (in Russian) (English transl.: Dokl. Phys. 53(11), 600-604 (2008)) · Zbl 1037.35051 [29] Nazarov, S.A.: Gap detection in the spectrum of an elastic periodic waveguide with a free surface. Zh. Vychisl. Mat. i Mat. Fiz. 49(2), 323-333 (2009). (in Russian) (English transl.: Comput. Math. Math. Phys. 49(2), 332-343 (2009).) · Zbl 1302.76028 [30] Nazarov, S.A.: Opening a gap in the essential spectrum of the elasticity problem in a periodic half-layer. Algebra Anal. 21(2), 166-202 (2009). (in Russian) (English transl.: St. Petersburg Math. J. 21, 2 (2009)) · Zbl 0924.35087 [31] Nazarov, S.A.: Asymptotic Theory of Thin Plates and Rods. Reduction of Dimension and Integral Estimates. Nauchnaya Kniga, Novosibirsk (2002). (in Russian) [32] Nazarov, S.A., Plamenevskii, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin (1994) [33] Nazarov, SA; Ruotsalainen, K; Taskinen, J, Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps, Appl. Anal., 89, 109-124, (2010) · Zbl 1186.35123 [34] Nazarov, SA; Taskinen, J, Spectral gaps for periodic piezoelectric waveguides, Z. Angew. Math. Phys., 66, 3017-3047, (2015) · Zbl 1333.35275 [35] Visik, MI; Ljusternik, LA, Regular degeneration and boundary layer of linear differential equations with small parameter, Am. Math. Soc. Transl., 20, 239-364, (1962) [36] Zhikov, V.: Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients. Algebra Anal. 16(5), 34-58 (2004). (in Russian) (English transl.: St. Petersburg Math. J. 16(5), 773-790) (2005).) · Zbl 1186.35123
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.