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Homogenization of systems with equi-integrable coefficients

Published online by Cambridge University Press:  08 August 2014

Marc Briane  and
Juan Casado-Díaz
Marc Briane
Affiliation:
Institut de Recherche Mathématique de Rennes, INSA de Rennes, France. mbriane@insa-rennes.fr
Juan Casado-Díaz
Affiliation:
Departemento. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain; jcasadod@us.es
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Abstract

In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for scalar equations does not hold for vector-valued systems, we present an alternative approach based on an approximation result by Lipschitz functions due to Acerbi and Fusco combined with a Meyers Lp-estimate adapted to the functional ellipticity condition. The present framework includes in particular the elasticity case and the reinforcement by stiff thin fibers.

Keywords

Homogenizationvector-valued systemsnot equi-bounded coefficientsequi-integrable coefficientsH-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1214 - 1223
Copyright
© EDP Sciences, SMAI, 2014

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References

E. Acerbi and N. Fusco, An approximation lemma for W 1, p functions, Material instabilities in continuum mechanics (Edinburgh, 1985-1986). Oxford Sci. Publ., Oxford Univ. Press, New York (1988) 1–5.
Acerbi, E. and Fusco, N., “A regularity theorem for minimizers of quasiconvex integrals”. Arch. Rational Mech. Anal. 99 (1987) 261–281. Google Scholar
Bellieud, M. and Bouchitté, G., Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998) 407–436. Google Scholar
Bellieud, M. and Gruais, I., 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. Google Scholar
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, corrected reprint of the 1978 original. AMS Chelsea Publishing, Providence (2011).
Beurling, A. and Deny, J., Espaces de Dirichlet. Acta Math. 99 (1958) 203–224. Google Scholar
A. Braides, Γ–convergence for Beginners. Oxford University Press, Oxford (2002).
Braides, A., Briane, M., and Casado-Díaz, J., Homogenization of non-uniformly bounded periodic diffusion energies in dimension two. Nonlinearity 22 (2009) 1459–1480. Google Scholar
Briane, M., Homogenization of high-conductivity periodic problems: Application to a general distribution of one-directional fibers. SIAM J. Math. Anal. 35 (2003) 33–60. Google Scholar
Briane, M. and Camar–Eddine, M., Homogenization of two-dimensional elasticity problems with very stiff coefficients. J. Math. Pures Appl. 88 (2007) 483–505. Google Scholar
Briane, M. and Camar–Eddine, M., An optimal condition of compactness for elasticity problems involving one directional reinforcement. J. Elasticity 107 (2012) 11–38. Google Scholar
Briane, M. and Casado–Díaz, J., Asymptotic behavior of equicoercive diffusion energies in two dimension. Calc. Var. Partial Differ. Equ. 29 (2007) 455–479. Google Scholar
Briane, M. and Casado–Díaz, J., Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects. Calc. Var. Partial Differ. Equ. 33 (2008) 463–492. Google Scholar
M. Briane and J. Casado–Díaz, A new div-curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian. In preparation.
Briane, M. and Tchou, N., Fibered microstructures for some nonlocal Dirichlet forms. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30 (2001) 681–711. Google Scholar
Camar–Eddine, M. and Seppecher, P., Closure of the set of diffusion functionals with respect to the Mosco-convergence. Math. Models Methods Appl. Sci. 12 (2002) 1153–1176. Google Scholar
Camar–Eddine, M. and Seppecher, P., Determination of the closure of the set of elasticity functionals. Arch. Ration. Mech. Anal. 170 (2003) 211–245. Google Scholar
Carbone, L. and Sbordone, C., Some properties of Γ-limits of integral functionals. Ann. Mat. Pura Appl. 122 (1979) 1–60. Google Scholar
G. Dal Maso, An introduction to Γ-convergence. Progr. Nonlin. Differ. Equ. Birkhaüser, Boston (1993).
De Giorgi, E., Sulla convergenza di alcune successioni di integrali del tipo dell’area. Rend. Mat. Appl. 8 (1975) 277–294. Google Scholar
De Giorgi, E., Γ-convergenza e G-convergenza. Boll. Un. Mat. Ital. 14-A (1977) 213–220. Google Scholar
N. Dunford and J.T. Schwartz, Linear operators. Part I. General theory. Wiley-Interscience Publication, New York (1988).
V.N. Fenchenko and E.Ya. Khruslov, Asymptotic of solution of differential equations with strongly oscillating matrix of coefficients which does not satisfy the condition of uniform boundedness. Dokl. AN Ukr. SSR 4 (1981).
E.Ya. Khruslov, Homogenized models of composite media, Composite Media and Homogenization Theory, edited by G. Dal Maso and G.F. Dell’Antonio, in Progr. Nonlin. Differ. Equ. Appl. Birkhaüser (1991) 159–182.
E.Ya. Khruslov and V.A. Marchenko, Homogenization of Partial Differential Equations, vol. 46. Progr. Math. Phys. Birkhäuser, Boston (2006).
Meyers, N.G., An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1963) 189–206. Google Scholar
Mosco, U., Composite media and asymptotic Dirichlet forms. J. Func. Anal. 123 (1994) 368–421. Google Scholar
F. Murat, H-convergence, Séminaire d’Analyse Fonctionnelle et Numérique, 1977-78, Université d’Alger, multicopied, 34 pp. English translation: F. Murat and L. Tartar, H-convergence. Topics in the Mathematical Modelling of Composite Materials, edited by L. Cherkaev and R.V. Kohn, vol. 31. Progr. Nonlin. Differ. Equ. Appl. Birkaüser, Boston (1998) 21–43.
Pideri, C. and Seppecher, P., A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9 (1997) 241–257. Google Scholar
Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1968) 571–597. Google Scholar
L. Tartar, The General Theory of Homogenization: A Personalized Introduction. Lect. Notes Unione Matematica Italiana. Springer-Verlag, Berlin, Heidelberg (2009).