doi: 10.3934/nhm.2021012

Combined effects of homogenization and singular perturbations: A bloch wave approach

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, Karnataka, 560065, India

Received  December 2020 Published  May 2021

In this work, we study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit. The main difficulty is the presence of two parameters which requires us to obtain uniform bounds on the Bloch spectral data in various regimes of the parameter.

Citation: Vivek Tewary. Combined effects of homogenization and singular perturbations: A bloch wave approach. Networks & Heterogeneous Media, doi: 10.3934/nhm.2021012
References:
[1]

E. C. Aifantis, On the Microstructural Origin of Certain Inelastic Models, Journal of Engineering Materials and Technology, 106 (1984), 326-330.  doi: 10.1115/1.3225725.  Google Scholar

[2]

G. AllaireM. Briane and M. Vanninathan, A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures, SeMA Journal. Boletin de la Sociedad Espanñola de Matemática Aplicada, 73 (2016), 237-259.  doi: 10.1007/s40324-016-0067-z.  Google Scholar

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G. AllaireT. Ghosh and M. Vanninathan, Homogenization of Stokes system using Bloch waves, Networks and Heterogeneous Media, 12 (2017), 525-550.  doi: 10.3934/nhm.2017022.  Google Scholar

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H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Operator Theory: Advances and Applications, Vol. 15, Birkhäuser Verlag, Basel, 1985.  Google Scholar

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A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.  Google Scholar

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M. Birman and T. Suslina, Periodic second-order differential operators. Threshold properties and homogenization, St. Petersburg Mathematical Journal, 15 (2004), 639-714.  doi: 10.1090/S1061-0022-04-00827-1.  Google Scholar

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A. Braides and C. I. Zeppieri, Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy, Interfaces and Free Boundaries, 11 (2009), 61-118.  doi: 10.4171/IFB/204.  Google Scholar

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L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates, Communications in Mathematical Physics, 303 (2011), 421-449.  doi: 10.1007/s00220-010-1171-z.  Google Scholar

[10]

C. ConcaD. GómezM. Lobo and E. Pérez, The Bloch approximation in periodically perforated media, Applied Mathematics and Optimization, 52 (2005), 93-127.  doi: 10.1007/s00245-005-0822-5.  Google Scholar

[11]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM Journal on Mathematical Analysis, 33 (2002), 1166-1198.  doi: 10.1137/S0036141001382200.  Google Scholar

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C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization on bounded domains, Asymptotic Analysis, 41 (2005), 71-91.   Google Scholar

[13]

C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM Journal on Applied Mathematics, 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.  Google Scholar

[14]

T. DohnalA. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014), 488-513.  doi: 10.1137/130935033.  Google Scholar

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T. DohnalA. Lamacz and B. Schweizer, Dispersive homogenized models and coefficient formulas for waves in general periodic media, Asymptotic Analysis, 93 (2015), 21-49.  doi: 10.3233/ASY-141280.  Google Scholar

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D. DupuyR. Orive and L. Smaranda, Bloch waves homogenization of a Dirichlet problem in a periodically perforated domain, Asymptotic Analysis, 61 (2009), 229-250.  doi: 10.3233/ASY-2008-0912.  Google Scholar

[17]

F. Ferraresso and P. D. Lamberti, On a Babuška paradox for polyharmonic operators: Spectral stability and boundary homogenization for intermediate problems, Integral Equations and Operator Theory, 91 (2019), Paper No. 55, 42 pp. doi: 10.1007/s00020-019-2552-0.  Google Scholar

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G. Franfort and S. Müller, Combined effects of homogenization and singular perturbations in elasticity, Journal für die reine und angewandte Mathematik (Crelles Journal), 454 (1994), 1-35.  doi: 10.1515/crll.1994.454.1.  Google Scholar

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S. S. Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptotic Analysis, 39 (2004), 15-44.   Google Scholar

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F. Gazzola, H. -C. Grunau and G. Sweers, Eigenvalue problems, in Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, Vol. 1991, Springer-Verlag, Berlin, 2010, 61-98. doi: 10.1007/978-3-642-12245-3.  Google Scholar

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T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.  Google Scholar

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A. Lamacz-Keymling and I. Yousept, High-order homogenization in optimal control by the Bloch wave method, preprint, 2020, arXiv: 2010.04469v1. Google Scholar

[23]

Y. K. Lee, Conditions for shear banding and material instability in finite elastoplastic deformation, International Journal of Plasticity, 5 (1989), 197-226.  doi: 10.1016/0749-6419(89)90013-2.  Google Scholar

[24]

F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, Birkhäuser Boston, Boston, MA, 1997, 21-43. doi: 10.1007/978-1-4612-2032-9.  Google Scholar

[25] R. Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics, University of Chicago Press, 1971.   Google Scholar
[26]

W. Niu and Z. Shen, Combined Effects of Homogenization and Singular Perturbations: Quantitative Estimates, preprint, 2020, arXiv: 2005.12776v1. Google Scholar

[27]

W. Niu and Y. Yuan, Convergence rate in homogenization of elliptic systems with singular perturbations, Journal of Mathematical Physics, 60 (2019), 111509, 7 pp. doi: 10.1063/1.5124140.  Google Scholar

[28]

J. OrtegaJ. S. Martín and L. Smaranda, Bloch wave homogenization of a non-homogeneous Neumann problem, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 969-993.  doi: 10.1007/s00033-007-6142-7.  Google Scholar

[29]

S. E. Pastukhova, Homogenization estimates for singularly perturbed operators, Journal of Mathematical Sciences, 251 (2020), 724-747.  doi: 10.1007/s10958-020-05125-0.  Google Scholar

[30]

L. Provenzano, A note on the Neumann eigenvalues of the biharmonic operator, Mathematical Methods in the Applied Sciences, 41 (2018), 1005-1012.  doi: 10.1002/mma.4063.  Google Scholar

[31]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar

[32]

Y. C. Shu, Heterogeneous thin films of Martensitic materials, Archive for Rational Mechanics and Analysis, 153 (2000), 39-90.  doi: 10.1007/s002050000088.  Google Scholar

[33]

S. Sivaji Ganesh and V. Tewary, Bloch approach to almost periodic homogenization and approximations of effective coefficients, Discrete and Continuous Dynamical Systems Series B, 2021, 1-36. doi: 10.3934/dcdsb. 2021119.  Google Scholar

[34]

S. Sivaji Ganesh and V. Tewary, Bloch wave homogenisation of quasiperiodic media, European Journal of Applied Mathematics, 2020, 1-21. doi: 10.1017/S0956792520000352.  Google Scholar

[35]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of linear elasticity system, ESAIM Control Optim. Calc. Var., 11 (2005), 542-573.  doi: 10.1051/cocv:2005018.  Google Scholar

[36]

T. Suslina, Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients, St. Petersburg Mathematical Journal, 29 (2018), 325-362.  doi: 10.1090/spmj/1496.  Google Scholar

[37]

N. Veniaminov, Homogenization of periodic differential operators of high order, St. Petersburg Mathematical Journal, 22 (2011), 751-775.  doi: 10.1090/S1061-0022-2011-01166-5.  Google Scholar

[38]

C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167.  doi: 10.1007/BF02790171.  Google Scholar

[39]

C. I. Zeppieri, Stochastic homogenisation of singularly perturbed integral functionals, Annali di Matematica Pura ed Applicata (4), 195 (2016), 2183-2208.  doi: 10.1007/s10231-016-0558-7.  Google Scholar

show all references

References:
[1]

E. C. Aifantis, On the Microstructural Origin of Certain Inelastic Models, Journal of Engineering Materials and Technology, 106 (1984), 326-330.  doi: 10.1115/1.3225725.  Google Scholar

[2]

G. AllaireM. Briane and M. Vanninathan, A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures, SeMA Journal. Boletin de la Sociedad Espanñola de Matemática Aplicada, 73 (2016), 237-259.  doi: 10.1007/s40324-016-0067-z.  Google Scholar

[3]

G. AllaireT. Ghosh and M. Vanninathan, Homogenization of Stokes system using Bloch waves, Networks and Heterogeneous Media, 12 (2017), 525-550.  doi: 10.3934/nhm.2017022.  Google Scholar

[4]

S. Bair, Chapter Ten - Shear localization, slip, and the limiting stress, in High Pressure Rheology for Quantitative Elastohydrodynamics second edition, Elsevier, 2019, 271-286. doi: 10.1016/B978-0-444-64156-4.00010-6.  Google Scholar

[5]

H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Operator Theory: Advances and Applications, Vol. 15, Birkhäuser Verlag, Basel, 1985.  Google Scholar

[6]

A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.  Google Scholar

[7]

M. Birman and T. Suslina, Periodic second-order differential operators. Threshold properties and homogenization, St. Petersburg Mathematical Journal, 15 (2004), 639-714.  doi: 10.1090/S1061-0022-04-00827-1.  Google Scholar

[8]

A. Braides and C. I. Zeppieri, Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy, Interfaces and Free Boundaries, 11 (2009), 61-118.  doi: 10.4171/IFB/204.  Google Scholar

[9]

L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates, Communications in Mathematical Physics, 303 (2011), 421-449.  doi: 10.1007/s00220-010-1171-z.  Google Scholar

[10]

C. ConcaD. GómezM. Lobo and E. Pérez, The Bloch approximation in periodically perforated media, Applied Mathematics and Optimization, 52 (2005), 93-127.  doi: 10.1007/s00245-005-0822-5.  Google Scholar

[11]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM Journal on Mathematical Analysis, 33 (2002), 1166-1198.  doi: 10.1137/S0036141001382200.  Google Scholar

[12]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization on bounded domains, Asymptotic Analysis, 41 (2005), 71-91.   Google Scholar

[13]

C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM Journal on Applied Mathematics, 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.  Google Scholar

[14]

T. DohnalA. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014), 488-513.  doi: 10.1137/130935033.  Google Scholar

[15]

T. DohnalA. Lamacz and B. Schweizer, Dispersive homogenized models and coefficient formulas for waves in general periodic media, Asymptotic Analysis, 93 (2015), 21-49.  doi: 10.3233/ASY-141280.  Google Scholar

[16]

D. DupuyR. Orive and L. Smaranda, Bloch waves homogenization of a Dirichlet problem in a periodically perforated domain, Asymptotic Analysis, 61 (2009), 229-250.  doi: 10.3233/ASY-2008-0912.  Google Scholar

[17]

F. Ferraresso and P. D. Lamberti, On a Babuška paradox for polyharmonic operators: Spectral stability and boundary homogenization for intermediate problems, Integral Equations and Operator Theory, 91 (2019), Paper No. 55, 42 pp. doi: 10.1007/s00020-019-2552-0.  Google Scholar

[18]

G. Franfort and S. Müller, Combined effects of homogenization and singular perturbations in elasticity, Journal für die reine und angewandte Mathematik (Crelles Journal), 454 (1994), 1-35.  doi: 10.1515/crll.1994.454.1.  Google Scholar

[19]

S. S. Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptotic Analysis, 39 (2004), 15-44.   Google Scholar

[20]

F. Gazzola, H. -C. Grunau and G. Sweers, Eigenvalue problems, in Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, Vol. 1991, Springer-Verlag, Berlin, 2010, 61-98. doi: 10.1007/978-3-642-12245-3.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[22]

A. Lamacz-Keymling and I. Yousept, High-order homogenization in optimal control by the Bloch wave method, preprint, 2020, arXiv: 2010.04469v1. Google Scholar

[23]

Y. K. Lee, Conditions for shear banding and material instability in finite elastoplastic deformation, International Journal of Plasticity, 5 (1989), 197-226.  doi: 10.1016/0749-6419(89)90013-2.  Google Scholar

[24]

F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, Birkhäuser Boston, Boston, MA, 1997, 21-43. doi: 10.1007/978-1-4612-2032-9.  Google Scholar

[25] R. Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics, University of Chicago Press, 1971.   Google Scholar
[26]

W. Niu and Z. Shen, Combined Effects of Homogenization and Singular Perturbations: Quantitative Estimates, preprint, 2020, arXiv: 2005.12776v1. Google Scholar

[27]

W. Niu and Y. Yuan, Convergence rate in homogenization of elliptic systems with singular perturbations, Journal of Mathematical Physics, 60 (2019), 111509, 7 pp. doi: 10.1063/1.5124140.  Google Scholar

[28]

J. OrtegaJ. S. Martín and L. Smaranda, Bloch wave homogenization of a non-homogeneous Neumann problem, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 969-993.  doi: 10.1007/s00033-007-6142-7.  Google Scholar

[29]

S. E. Pastukhova, Homogenization estimates for singularly perturbed operators, Journal of Mathematical Sciences, 251 (2020), 724-747.  doi: 10.1007/s10958-020-05125-0.  Google Scholar

[30]

L. Provenzano, A note on the Neumann eigenvalues of the biharmonic operator, Mathematical Methods in the Applied Sciences, 41 (2018), 1005-1012.  doi: 10.1002/mma.4063.  Google Scholar

[31]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar

[32]

Y. C. Shu, Heterogeneous thin films of Martensitic materials, Archive for Rational Mechanics and Analysis, 153 (2000), 39-90.  doi: 10.1007/s002050000088.  Google Scholar

[33]

S. Sivaji Ganesh and V. Tewary, Bloch approach to almost periodic homogenization and approximations of effective coefficients, Discrete and Continuous Dynamical Systems Series B, 2021, 1-36. doi: 10.3934/dcdsb. 2021119.  Google Scholar

[34]

S. Sivaji Ganesh and V. Tewary, Bloch wave homogenisation of quasiperiodic media, European Journal of Applied Mathematics, 2020, 1-21. doi: 10.1017/S0956792520000352.  Google Scholar

[35]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of linear elasticity system, ESAIM Control Optim. Calc. Var., 11 (2005), 542-573.  doi: 10.1051/cocv:2005018.  Google Scholar

[36]

T. Suslina, Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients, St. Petersburg Mathematical Journal, 29 (2018), 325-362.  doi: 10.1090/spmj/1496.  Google Scholar

[37]

N. Veniaminov, Homogenization of periodic differential operators of high order, St. Petersburg Mathematical Journal, 22 (2011), 751-775.  doi: 10.1090/S1061-0022-2011-01166-5.  Google Scholar

[38]

C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167.  doi: 10.1007/BF02790171.  Google Scholar

[39]

C. I. Zeppieri, Stochastic homogenisation of singularly perturbed integral functionals, Annali di Matematica Pura ed Applicata (4), 195 (2016), 2183-2208.  doi: 10.1007/s10231-016-0558-7.  Google Scholar

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