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

Michel Bellieud

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 Ø × ( 0 , T ) + boundary conditions , n { 1 , 2 } , when the coefficients ρ ε , a ε (defined in Ω) 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 (2010): 266-284. <http://eudml.org/doc/90765>.

@article{Bellieud2010,
abstract = { We study the homogenization of parabolic or hyperbolic equations like \[ \rho\_\varepsilon\{\partial^n u\_\varepsilon\over \partial t^n\}- \{\rm div\}( a\_\varepsilon\nabla u\_\varepsilon) =f \ \ \hbox\{ in \} \quad \{\O\times(0,T)\}+ \ \ \hbox\{\rm boundary conditions\}, \quad n \in \\{1,2\\}, \] when the coefficients $\rho_\varepsilon$, $a_\varepsilon$ (defined in Ω) take possibly high values on a ε-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; grain-like inclusions},
language = {eng},
month = {3},
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/90765},
volume = {11},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 2
SP - 266
EP - 284
AB - We study the homogenization of parabolic or hyperbolic equations like \[ \rho_\varepsilon{\partial^n u_\varepsilon\over \partial t^n}- {\rm div}( a_\varepsilon\nabla u_\varepsilon) =f \ \ \hbox{ in } \quad {\O\times(0,T)}+ \ \ \hbox{\rm boundary conditions}, \quad n \in \{1,2\}, \] when the coefficients $\rho_\varepsilon$, $a_\varepsilon$ (defined in Ω) take possibly high values on a ε-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; grain-like inclusions
UR - http://eudml.org/doc/90765
ER -

References

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  1. M. Bellieud, Homogenization of evolution problems in a fiber reinforced structure. J. Convex Anal.11 (2004) 363–385.  Zbl1071.35011
  2. M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Non local effects. Ann. Scuola Norm. Sup. Cl. Sci. IV26 (1998) 407–436.  Zbl0919.35014
  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 mécanique337 (2003) 493–498.  Zbl1027.35011
  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. H. Brezis, Analyse fonctionnelle. Masson, Paris (1983).  Zbl0511.46001
  6. G. Dal Maso, An introduction to Γ -Convergence. Progress Nonlinear Differential Equations Appl., Birkhäuser, Boston (1993).  Zbl0816.49001
  7. E.Y. Khruslov, Homogenized models of composite media. Progress Nonlinear Differential Equations Appl., Birkhäuser (1991).  Zbl0737.73009
  8. J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod, Paris 1 (1968).  Zbl0165.10801
  9. U. Mosco, Composite media and asymptotic Dirichlet forms. J. Funct. Anal.123 (1994) 368–421.  Zbl0808.46042
  10. G. Panasenko, Multicomponent homogenization of the vibration problem for incompressible media with heavy and rigid inclusions. C. R. Acad. Sci. Paris I321 (1995) 1109–1114.  Zbl0840.73005

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