Homogenization of evolution problems for a composite medium with very small and heavy inclusions

Michel Bellieud

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 2, page 266-284
  • ISSN: 1292-8119

Abstract

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We study the homogenization of parabolic or hyperbolic equations like ρ ε ∂ n u ε ∂ t n - div ( a ε ∇ u ε ) = f in Ω × ( 0 , T ) + boundary conditions , n ∈ { 1 , 2 } , when the coefficients ρ ε , a ε (defined in Ø ) take possibly high values on a ε -periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

How to cite

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Bellieud, Michel. "Homogenization of evolution problems for a composite medium with very small and heavy inclusions." ESAIM: Control, Optimisation and Calculus of Variations 11.2 (2005): 266-284. <http://eudml.org/doc/244688>.

@article{Bellieud2005,
abstract = {We study the homogenization of parabolic or hyperbolic equations like\[ \rho \_\varepsilon \{\partial ^n u\_\varepsilon \over \partial t^n\}- \{\rm div\}( a\_\varepsilon \nabla u\_\varepsilon ) =f \ \ \text\{in\} \quad \{\Omega \times (0,T)\}+ \ \ \text\{boundary\} \text\{conditions\}, \quad n \in \lbrace 1,2\rbrace , \]when the coefficients $\rho _\varepsilon $, $a_\varepsilon $ (defined in $Ø$) take possibly high values on a $\varepsilon $-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.},
author = {Bellieud, Michel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {homogenization; memory effects; grain-like inclusions; limit problem},
language = {eng},
number = {2},
pages = {266-284},
publisher = {EDP-Sciences},
title = {Homogenization of evolution problems for a composite medium with very small and heavy inclusions},
url = {http://eudml.org/doc/244688},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Bellieud, Michel
TI - Homogenization of evolution problems for a composite medium with very small and heavy inclusions
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 2
SP - 266
EP - 284
AB - We study the homogenization of parabolic or hyperbolic equations like\[ \rho _\varepsilon {\partial ^n u_\varepsilon \over \partial t^n}- {\rm div}( a_\varepsilon \nabla u_\varepsilon ) =f \ \ \text{in} \quad {\Omega \times (0,T)}+ \ \ \text{boundary} \text{conditions}, \quad n \in \lbrace 1,2\rbrace , \]when the coefficients $\rho _\varepsilon $, $a_\varepsilon $ (defined in $Ø$) take possibly high values on a $\varepsilon $-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.
LA - eng
KW - homogenization; memory effects; grain-like inclusions; limit problem
UR - http://eudml.org/doc/244688
ER -

References

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  1. [1] M. Bellieud, Homogenization of evolution problems in a fiber reinforced structure. J. Convex Anal. 11 (2004) 363–385. Zbl1071.35011
  2. [2] M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Non local effects. Ann. Scuola Norm. Sup. Cl. Sci. IV 26 (1998) 407–436. Zbl0919.35014
  3. [3] M. Bellieud and I. Gruais, Homogénéisation d’une structure élastique renforcée de fibres très rigides. Effets non locaux. C. R. Math., Problèmes mathématiques de la mécanique 337 (2003) 493–498. Zbl1027.35011
  4. [4] M. Bellieud and I. Gruais, Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non local effects. Memory effects. J. Math. Pures Appl. 84 (2005) 55–96. Zbl1079.74052
  5. [5] H. Brezis, Analyse fonctionnelle. Masson, Paris (1983). Zbl0511.46001MR697382
  6. [6] G. Dal Maso, An introduction to Γ -Convergence. Progress Nonlinear Differential Equations Appl., Birkhäuser, Boston (1993). Zbl0816.49001MR1201152
  7. [7] E.Y. Khruslov, Homogenized models of composite media. Progress Nonlinear Differential Equations Appl., Birkhäuser (1991). Zbl0737.73009MR1145750
  8. [8] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod, Paris 1 (1968). Zbl0165.10801
  9. [9] U. Mosco, Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 (1994) 368–421. Zbl0808.46042
  10. [10] G. Panasenko, Multicomponent homogenization of the vibration problem for incompressible media with heavy and rigid inclusions. C. R. Acad. Sci. Paris I 321 (1995) 1109–1114. Zbl0840.73005

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