American Institute of Mathematical Sciences

Advanced
September  2008, 3(3): 413-436. doi: 10.3934/nhm.2008.3.413

Homogenization of spectral problems in bounded domains with doubly high contrasts

Natalia O. Babych 1, , Ilia V. Kamotski 1, and  Valery P. Smyshlyaev 1,

1. 

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom, United Kingdom, United Kingdom

Received  April 2008 Published  June 2008

Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain non-standard self-adjoint operators. In particular they explicitly display the first two terms in the asymptotic expansion for the eigenvalues, with a surprising bound for the error of order $\varepsilon^{5/4}$ proved.
Keywords: high-contrasts, Homogenization, eigenvalue asymptotics, periodic media.
Mathematics Subject Classification: Primary: 35B27; Secondary: 34.
Citation: Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev. Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks & Heterogeneous Media, 2008, 3 (3) : 413-436. doi: 10.3934/nhm.2008.3.413
[1]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[2]

Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473
[3]

François Murat, Ali Sili. A remark about the periodic homogenization of certain composite fibered media. Networks & Heterogeneous Media, 2020, 15 (1) : 125-142. doi: 10.3934/nhm.2020006
[4]

Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29
[5]

Mohamed Camar-Eddine, Laurent Pater. Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures. Networks & Heterogeneous Media, 2013, 8 (4) : 913-941. doi: 10.3934/nhm.2013.8.913
[6]

Eugenia Pérez. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 859-883. doi: 10.3934/dcdsb.2007.7.859
[7]

Antonin Chambolle, Gilles Thouroude. Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem. Networks & Heterogeneous Media, 2009, 4 (1) : 127-152. doi: 10.3934/nhm.2009.4.127
[8]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[9]

Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91
[10]

María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks & Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004
[11]

Naoufel Ben Abdallah, Hédia Chaker. Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation. Kinetic & Related Models, 2009, 2 (3) : 403-424. doi: 10.3934/krm.2009.2.403
[12]

Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic & Related Models, 2015, 8 (2) : 201-214. doi: 10.3934/krm.2015.8.201
[13]

João Fialho, Feliz Minhós. High order periodic impulsive problems. Conference Publications, 2015, 2015 (special) : 446-454. doi: 10.3934/proc.2015.0446
[14]

Laura Sigalotti. Homogenization of pinning conditions on periodic networks. Networks & Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543
[15]

Marc Briane, David Manceau. Duality results in the homogenization of two-dimensional high-contrast conductivities. Networks & Heterogeneous Media, 2008, 3 (3) : 509-522. doi: 10.3934/nhm.2008.3.509
[16]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[17]

Léo Girardin. Competition in periodic media:Ⅰ-Existence of pulsating fronts. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1341-1360. doi: 10.3934/dcdsb.2017065
[18]

Ben Schweizer, Marco Veneroni. The needle problem approach to non-periodic homogenization. Networks & Heterogeneous Media, 2011, 6 (4) : 755-781. doi: 10.3934/nhm.2011.6.755
[19]

Michel Lenczner. Homogenization of linear spatially periodic electronic circuits. Networks & Heterogeneous Media, 2006, 1 (3) : 467-494. doi: 10.3934/nhm.2006.1.467
[20]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences