The band-gap structure of the spectrum in a periodic medium of masonry type

Günter Leugering; Sergei A. Nazarov; Jari Taskinen

doi:10.3934/nhm.2020014

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December 2020, 15(4): 555-580. doi: 10.3934/nhm.2020014

The band-gap structure of the spectrum in a periodic medium of masonry type

Günter Leugering ^1, , Sergei A. Nazarov ^2, and Jari Taskinen ^3,

1.	Department Mathematik, Lehrstuhl für Angewandte Mathematik 2, Cauerstr. 11, 91058 Erlangen, Germany
2.	Saint-Petersburg State University, Universitetskaya nab. 7–9, St. Petersburg, 199034, Russia, and, Institute for Problems in Mechanical Engineering of RAS, St. Petersburg, 199178, Russia
3.	Department of Mathematics and Statistics, University of Helsinki, P.O.Box 68, 00014 Helsinki, Finland

Received August 2019 Revised April 2020 Published December 2020 Early access July 2020

Fund Project: The second named author was supported by the Russian Foundation on Basic Research, project 18-01-00325

We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane $ \mathbb{R}^2 $. The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane $ \mathbb{R}^2 $ is decomposed into an infinite union of the translates of the rectangular periodicity cell $ \Omega^0 $, and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of $ \Omega^0 $ consist of a neighborhood of the boundary of the cell of the width $ h $ and thus has an area comparable to $ h $, where $ h>0 $ is a small parameter.

Using the methods of asymptotic analysis we study the position of the spectral bands as $ h \to 0 $ and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided $ h $ is small enough.

Keywords: Second order elliptic system, essential spectrum, periodic medium, band-gap spectrum, spectral gap.

Mathematics Subject Classification: Primary: 35J57; Secondary: 35P99, 47B25.

Citation: Günter Leugering, Sergei A. Nazarov, Jari Taskinen. The band-gap structure of the spectrum in a periodic medium of masonry type. Networks & Heterogeneous Media, 2020, 15 (4) : 555-580. doi: 10.3934/nhm.2020014

References:

[1]	F. L. Bakharev, G. Cardone, S. A. Nazarov and J. Taskinen, Effects of Rayleigh waves on the essential spectrum in perturbed doubly periodic elliptic problems, Integral Equations Operator Theory, 88 (2017), 373-386. doi: 10.1007/s00020-017-2379-5. Google Scholar
[2]	F. L. Bakharev and J. Taskinen, Bands in the spectrum of a periodic elastic waveguide, Z. Angew. Math. Phys., 68 (2017), 27 pp. doi: 10.1007/s00033-017-0846-0. Google Scholar
[3]	M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987. Google Scholar
[4]	G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal., 50 (2006), 121-173. Google Scholar
[5]	I. M. Gelfand, Expansion in characteristic functions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR, 73 (1950), 1117-1120. Google Scholar
[6]	R. Hempel and K. Lienau, Spectral properties of periodic media in the large coupling limit, Comm. Partial Differential Equations, 25 (2000), 1445-1470. doi: 10.1080/03605300008821555. Google Scholar
[7]	R. Hempel and O. Post, Spectral gaps for periodic elliptic operators with high contrast: An overview, Progress in analysis, Vol. I, II (Berlin, 2001), 577–587. Google Scholar
[8]	V. A. Kondratev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 16 (1967), 209-292. Google Scholar
[9]	V. A. Kozlov and V. G. Maz'ya, Spectral properties of the operator bundles generated by elliptic boundary-value problems in a cone, translation in Funct. Anal. Appl., 22 (1988), 114-121. doi: 10.1007/BF01077601. Google Scholar
[10]	V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains With Point Singularities, Amer. Math. Soc., Providence RI, 1997. Google Scholar
[11]	P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk, 37 (1982), 3-52. Google Scholar
[12]	P. A. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8573-7. Google Scholar
[13]	S. Langer, S. A. Nazarov and M. Specovius-Neugebauer, Artificial boundary conditions on polyhedral truncation surfaces for three-dimensional elasticity systems, Comptes Rendus Mécanique, 332 (2004), 591-596. doi: 10.1016/j.crme.2004.03.011. Google Scholar
[14]	J.–L. Lions and E. Magenes, Non-Homogeneus Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[15]	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. Nachr., 76 (1977), 29-60. doi: 10.1002/mana.19770760103. Google Scholar
[16]	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 (1978), 25-82. doi: 10.1002/mana.19780810103. Google Scholar
[17]	S. A. Nazarov, The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes, translation in Russian Math. Surveys, 54 (1999), 947-1014. doi: 10.1070/rm1999v054n05ABEH000204. Google Scholar
[18]	S. A. Nazarov, Asymptotic behavior of the solution and the modeling of the Dirichlet problem in an angular domain with rapidly oscillating boundary, translation in St. Petersburg Math. J., 19 (2008), 297-326. doi: 10.1090/S1061-0022-08-01000-5. Google Scholar
[19]	S. A. Nazarov, The Neumann problem in angular domains with periodic boundaries and parabolic perturbations of the boundaries, Tr. Mosk. Mat. Obs., 69(2008), 182–241; translation in Trans. Moscow Math. Soc., (2008), 153–208. doi: 10.1090/S0077-1554-08-00173-8. Google Scholar
[20]	S. A. Nazarov, Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equation, translation in Differ. Equ., 46 (2010), 730-741. doi: 10.1134/S0012266110050125. Google Scholar
[21]	S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110848915.525. Google Scholar
[22]	S. A. Nazarov, K. Ruotsalainen and J. Taskinen, Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps, Appl. Anal., 89 (2010), 109-124. doi: 10.1080/00036810903479715. Google Scholar
[23]	S. A. Nazarov and J. Taskinen, Spectral gaps for periodic piezoelectric waveguides, Z. Angew. Math. Phys., 66 (2015), 3017-3047. doi: 10.1007/s00033-015-0561-7. Google Scholar
[24]	J. Nečas, Les Méthodes in Théorie Des Équations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967. Google Scholar
[25]	M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov., 171 (1985), 122 pp. Google Scholar
[26]	M. Specovius-Neugebauer and M. Steigemann, Eigenfunctions of the 2-dimensional anisotropic elasticity operator and algebraic equivalent materials, ZAMM Z. Angew. Math. Mech., 88 (2008), 100-115. doi: 10.1002/zamm.200700086. Google Scholar
[27]	M. I. Vishik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspehi Mat. Nauk (N.S.), 12 1957, 3–122. Google Scholar
[28]	V. V. Zhikov, On gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients, translation in St. Petersburg Math. J., 16 (2005), 773-790. doi: 10.1090/S1061-0022-05-00878-2. Google Scholar

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References:

[1]	F. L. Bakharev, G. Cardone, S. A. Nazarov and J. Taskinen, Effects of Rayleigh waves on the essential spectrum in perturbed doubly periodic elliptic problems, Integral Equations Operator Theory, 88 (2017), 373-386. doi: 10.1007/s00020-017-2379-5. Google Scholar
[2]	F. L. Bakharev and J. Taskinen, Bands in the spectrum of a periodic elastic waveguide, Z. Angew. Math. Phys., 68 (2017), 27 pp. doi: 10.1007/s00033-017-0846-0. Google Scholar
[3]	M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987. Google Scholar
[4]	G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptot. Anal., 50 (2006), 121-173. Google Scholar
[5]	I. M. Gelfand, Expansion in characteristic functions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR, 73 (1950), 1117-1120. Google Scholar
[6]	R. Hempel and K. Lienau, Spectral properties of periodic media in the large coupling limit, Comm. Partial Differential Equations, 25 (2000), 1445-1470. doi: 10.1080/03605300008821555. Google Scholar
[7]	R. Hempel and O. Post, Spectral gaps for periodic elliptic operators with high contrast: An overview, Progress in analysis, Vol. I, II (Berlin, 2001), 577–587. Google Scholar
[8]	V. A. Kondratev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 16 (1967), 209-292. Google Scholar
[9]	V. A. Kozlov and V. G. Maz'ya, Spectral properties of the operator bundles generated by elliptic boundary-value problems in a cone, translation in Funct. Anal. Appl., 22 (1988), 114-121. doi: 10.1007/BF01077601. Google Scholar
[10]	V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains With Point Singularities, Amer. Math. Soc., Providence RI, 1997. Google Scholar
[11]	P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk, 37 (1982), 3-52. Google Scholar
[12]	P. A. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8573-7. Google Scholar
[13]	S. Langer, S. A. Nazarov and M. Specovius-Neugebauer, Artificial boundary conditions on polyhedral truncation surfaces for three-dimensional elasticity systems, Comptes Rendus Mécanique, 332 (2004), 591-596. doi: 10.1016/j.crme.2004.03.011. Google Scholar
[14]	J.–L. Lions and E. Magenes, Non-Homogeneus Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[15]	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. Nachr., 76 (1977), 29-60. doi: 10.1002/mana.19770760103. Google Scholar
[16]	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 (1978), 25-82. doi: 10.1002/mana.19780810103. Google Scholar
[17]	S. A. Nazarov, The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes, translation in Russian Math. Surveys, 54 (1999), 947-1014. doi: 10.1070/rm1999v054n05ABEH000204. Google Scholar
[18]	S. A. Nazarov, Asymptotic behavior of the solution and the modeling of the Dirichlet problem in an angular domain with rapidly oscillating boundary, translation in St. Petersburg Math. J., 19 (2008), 297-326. doi: 10.1090/S1061-0022-08-01000-5. Google Scholar
[19]	S. A. Nazarov, The Neumann problem in angular domains with periodic boundaries and parabolic perturbations of the boundaries, Tr. Mosk. Mat. Obs., 69(2008), 182–241; translation in Trans. Moscow Math. Soc., (2008), 153–208. doi: 10.1090/S0077-1554-08-00173-8. Google Scholar
[20]	S. A. Nazarov, Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equation, translation in Differ. Equ., 46 (2010), 730-741. doi: 10.1134/S0012266110050125. Google Scholar
[21]	S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110848915.525. Google Scholar
[22]	S. A. Nazarov, K. Ruotsalainen and J. Taskinen, Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps, Appl. Anal., 89 (2010), 109-124. doi: 10.1080/00036810903479715. Google Scholar
[23]	S. A. Nazarov and J. Taskinen, Spectral gaps for periodic piezoelectric waveguides, Z. Angew. Math. Phys., 66 (2015), 3017-3047. doi: 10.1007/s00033-015-0561-7. Google Scholar
[24]	J. Nečas, Les Méthodes in Théorie Des Équations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967. Google Scholar
[25]	M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov., 171 (1985), 122 pp. Google Scholar
[26]	M. Specovius-Neugebauer and M. Steigemann, Eigenfunctions of the 2-dimensional anisotropic elasticity operator and algebraic equivalent materials, ZAMM Z. Angew. Math. Mech., 88 (2008), 100-115. doi: 10.1002/zamm.200700086. Google Scholar
[27]	M. I. Vishik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspehi Mat. Nauk (N.S.), 12 1957, 3–122. Google Scholar
[28]	V. V. Zhikov, On gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients, translation in St. Petersburg Math. J., 16 (2005), 773-790. doi: 10.1090/S1061-0022-05-00878-2. Google Scholar

Figure 1. The original (a) and limit (b) periodicity cell

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Figure 2. Additional geometric objects

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Figure 3. Pavements of different shapes

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The band-gap structure of the spectrum in a periodic medium of masonry type

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