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  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.

Citation: Günter Leugering, Sergei A. Nazarov, Jari Taskinen. The band-gap structure of the spectrum in a periodic medium of masonry type. Networks and Heterogeneous Media, 2020, 15 (4) : 555-580. doi: 10.3934/nhm.2020014
References:
[1]

F. L. BakharevG. CardoneS. 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.

[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.

[3]

M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987.

[4]

G. CalozM. CostabelM. 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. 

[5]

I. M. Gelfand, Expansion in characteristic functions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR, 73 (1950), 1117-1120. 

[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.

[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.

[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. 

[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.

[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.

[11]

P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk, 37 (1982), 3-52. 

[12]

P. A. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8573-7.

[13]

S. LangerS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

S. A. NazarovK. 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.

[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.

[24]

J. Nečas, Les Méthodes in Théorie Des Équations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967.

[25]

M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov., 171 (1985), 122 pp.

[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.

[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.

[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.

show all references

References:
[1]

F. L. BakharevG. CardoneS. 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.

[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.

[3]

M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987.

[4]

G. CalozM. CostabelM. 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. 

[5]

I. M. Gelfand, Expansion in characteristic functions of an equation with periodic coefficients, Dokl. Akad. Nauk SSSR, 73 (1950), 1117-1120. 

[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.

[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.

[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. 

[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.

[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.

[11]

P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk, 37 (1982), 3-52. 

[12]

P. A. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8573-7.

[13]

S. LangerS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

S. A. NazarovK. 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.

[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.

[24]

J. Nečas, Les Méthodes in Théorie Des Équations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967.

[25]

M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov., 171 (1985), 122 pp.

[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.

[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.

[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.

Figure 1.  The original (a) and limit (b) periodicity cell
Figure 2.  Additional geometric objects
Figure 3.  Pavements of different shapes
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