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Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect (2018)
Journal Article
Cherdantsev, M., Cherednichenko, K., & Cooper, S. (2018). Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect. SIAM Journal on Mathematical Analysis, 50(6), 5825-5856. https://doi.org/10.1137/17m112261x

Following a number of recent studies of resolvent and spectral convergence of nonuniformly elliptic families of differential operators describing the behavior of periodic composite media with high contrast, we study the corresponding one-dimensional... Read More about Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect.

Asymptotic behaviour of the spectra of systems of Maxwell equations in periodic composite media with high contrast (2018)
Journal Article
Cherednichenko, K., & Cooper, S. (2018). Asymptotic behaviour of the spectra of systems of Maxwell equations in periodic composite media with high contrast. Mathematika, 64(2), 583-605. https://doi.org/10.1112/s0025579318000062

We analyse the behaviour of the spectrum of the system of Maxwell equations of electromagnetism, with rapidly oscillating periodic coefficients, subject to periodic boundary conditions on a “macroscopic” domain (0, T ) 3 , T > 0. We consider the case... Read More about Asymptotic behaviour of the spectra of systems of Maxwell equations in periodic composite media with high contrast.

Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media (2018)
Journal Article
Cooper, S. (2018). Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media. Calculus of Variations and Partial Differential Equations, 57(3), Article 76. https://doi.org/10.1007/s00526-018-1365-3

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (... Read More about Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media.

Asymptotic Analysis of Stratified Elastic Media in the Space of Functions with Bounded Deformation (2017)
Journal Article
Bellieud, M., & Cooper, S. (2017). Asymptotic Analysis of Stratified Elastic Media in the Space of Functions with Bounded Deformation. SIAM Journal on Mathematical Analysis, 49(5), 4275-4317. https://doi.org/10.1137/16m107551x

We consider a heterogeneous elastic structure which is stratied in one direction. We derive the limit problem under the sole assumption that the Lame coecients and their inverses weakly* converge to some Radon measures.

Analyse asymptotique de milieux élastiques stratifiés dans les espaces de fonctions à déformation bornée. = Asymptotic analysis of stratified elastic media in the space of functions with bounded deformation (2016)
Journal Article
Bellieud, M., & Cooper, S. (2016). Analyse asymptotique de milieux élastiques stratifiés dans les espaces de fonctions à déformation bornée. = Asymptotic analysis of stratified elastic media in the space of functions with bounded deformation. Comptes Rendus Mathématique, 354(4), 437-442. https://doi.org/10.1016/j.crma.2016.01.004

Nous analysons le comportement asymptotique des solutions de problèmes du type(1)(Pε):{−div(σε(uε))=f dans Ω=(0,L)×Ω′,σε(uε)=λε(x1)tr(e(uε))I+2με(x1)e(uε),e(uε)=12(∇uε+∇Tuε),uε∈H01(Ω;R3),f∈L∞(Ω,R3), lorsque les coefficients de Lamé dépendent uniqueme... Read More about Analyse asymptotique de milieux élastiques stratifiés dans les espaces de fonctions à déformation bornée. = Asymptotic analysis of stratified elastic media in the space of functions with bounded deformation.

Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients (2015)
Journal Article
Cherednichenko, K., & Cooper, S. (2016). Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients. Archive for Rational Mechanics and Analysis, 219(3), 1061-1086. https://doi.org/10.1007/s00205-015-0916-4

We study the asymptotic behaviour of the resolvents (Aε+I)−1 of elliptic second-order differential operators Aε in Rd with periodic rapidly oscillating coefficients, as the period ε goes to zero. The class of operators covered by our analysis include... Read More about Resolvent Estimates for High-Contrast Elliptic Problems with Periodic Coefficients.

On the existence of high-frequency boundary resonances in layered elastic media (2015)
Journal Article
Cherednichenko, K., & Cooper, S. (2015). On the existence of high-frequency boundary resonances in layered elastic media. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2178), Article 20140878. https://doi.org/10.1098/rspa.2014.0878

We analyse the asymptotic behaviour of high-frequency vibrations of a three-dimensional layered elastic medium occupying the domain Ω=(−a,a)3, a>0. We show that in both cases of stress-free and zero-displacement boundary conditions on the boundary of... Read More about On the existence of high-frequency boundary resonances in layered elastic media.

Homogenization of the system of high-contrast Maxwell equations (2015)
Journal Article
Cherednichenko, K., & Cooper, S. (2015). Homogenization of the system of high-contrast Maxwell equations. Mathematika, 61(02), 475-500. https://doi.org/10.1112/s0025579314000424

We study the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities ε have a high degree of contrast between each other. We assume that the ratio between the permittivities of the compon... Read More about Homogenization of the system of high-contrast Maxwell equations.

Spectral Analysis of One-Dimensional High-Contrast Elliptic Problems with Periodic Coefficients (2015)
Journal Article
Cherednichenko, K., Cooper, S., & Guenneau, S. (2015). Spectral Analysis of One-Dimensional High-Contrast Elliptic Problems with Periodic Coefficients. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 13(1), 72-98. https://doi.org/10.1137/130947106

We study the behavior of the spectrum of a family of one-dimensional operators with periodic high-contrast coefficients as the period goes to zero, which may represent, e.g., the elastic or electromagnetic response of a two-component composite medium... Read More about Spectral Analysis of One-Dimensional High-Contrast Elliptic Problems with Periodic Coefficients.