# American Institute of Mathematical Sciences

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

 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  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.

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

show all references

##### 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
The original (a) and limit (b) periodicity cell
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