Homogenization of systems with equi-integrable coefficients

Marc Briane; Juan Casado-Díaz

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 4, page 1214-1223
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Briane, Marc, and Casado-Díaz, Juan. "Homogenization of systems with equi-integrable coefficients." ESAIM: Control, Optimisation and Calculus of Variations 20.4 (2014): 1214-1223. <http://eudml.org/doc/272933>.

@article{Briane2014,
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.},
author = {Briane, Marc, Casado-Díaz, Juan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {homogenization; vector-valued systems; not equi-bounded coefficients; equi-integrable coefficients; H-convergence; Lusin approximation; Meyers inequality; homogeneous Dirichlet boundary conditions},
language = {eng},
number = {4},
pages = {1214-1223},
publisher = {EDP-Sciences},
title = {Homogenization of systems with equi-integrable coefficients},
url = {http://eudml.org/doc/272933},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Briane, Marc
AU - Casado-Díaz, Juan
TI - Homogenization of systems with equi-integrable coefficients
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 4
SP - 1214
EP - 1223
AB - 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.
LA - eng
KW - homogenization; vector-valued systems; not equi-bounded coefficients; equi-integrable coefficients; H-convergence; Lusin approximation; Meyers inequality; homogeneous Dirichlet boundary conditions
UR - http://eudml.org/doc/272933
ER -

References

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  1. [1] E. Acerbi and N. Fusco, An approximation lemma for W1, p functions, Material instabilities in continuum mechanics (Edinburgh, 1985-1986). Oxford Sci. Publ., Oxford Univ. Press, New York (1988) 1–5. Zbl0644.46026MR970512
  2. [2] E. Acerbi and N. Fusco, “A regularity theorem for minimizers of quasiconvex integrals”. Arch. Rational Mech. Anal.99 (1987) 261–281. Zbl0627.49007MR888453
  3. [3] M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998) 407–436. Zbl0919.35014MR1635769
  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. Zbl1079.74052MR2112872
  5. [5] A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, corrected reprint of the 1978 original. AMS Chelsea Publishing, Providence (2011). Zbl1229.35001MR503330
  6. [6] A. Beurling and J. Deny, Espaces de Dirichlet. Acta Math.99 (1958) 203–224. Zbl0089.08106MR98924
  7. [7] A. Braides, Γ–convergence for Beginners. Oxford University Press, Oxford (2002). Zbl1198.49001MR1968440
  8. [8] A. Braides, M. Briane, and J. Casado-Díaz, Homogenization of non-uniformly bounded periodic diffusion energies in dimension two. Nonlinearity22 (2009) 1459–1480. Zbl1221.35048MR2507329
  9. [9] M. Briane, Homogenization of high-conductivity periodic problems: Application to a general distribution of one-directional fibers. SIAM J. Math. Anal.35 (2003) 33–60. Zbl1055.35017MR2001464
  10. [10] M. Briane and M. Camar–Eddine, Homogenization of two-dimensional elasticity problems with very stiff coefficients. J. Math. Pures Appl.88 (2007) 483–505. Zbl1134.35010MR2373738
  11. [11] M. Briane and M. Camar–Eddine, An optimal condition of compactness for elasticity problems involving one directional reinforcement. J. Elasticity107 (2012) 11–38. Zbl06135617MR2870292
  12. [12] M. Briane and J. Casado–Díaz, Asymptotic behavior of equicoercive diffusion energies in two dimension. Calc. Var. Partial Differ. Equ.29 (2007) 455–479. Zbl1186.35012MR2329805
  13. [13] M. Briane and J. Casado–Díaz, 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. Zbl1167.35336MR2438743
  14. [14] 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. Zbl1336.35045
  15. [15] M. Briane and N. Tchou, Fibered microstructures for some nonlocal Dirichlet forms. Ann. Scuola Norm. Sup. Pisa Cl. Sci.30 (2001) 681–711. Zbl1170.35321MR1896082
  16. [16] M. Camar–Eddine and P. Seppecher, Closure of the set of diffusion functionals with respect to the Mosco-convergence. Math. Models Methods Appl. Sci.12 (2002) 1153–1176. Zbl1032.35033MR1924605
  17. [17] M. Camar–Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals. Arch. Ration. Mech. Anal.170 (2003) 211–245. Zbl1030.74013MR2020260
  18. [18] L. Carbone and C. Sbordone, Some properties of Γ-limits of integral functionals. Ann. Mat. Pura Appl.122 (1979) 1–60. Zbl0474.49016MR565062
  19. [19] G. Dal Maso, An introduction to Γ-convergence. Progr. Nonlin. Differ. Equ. Birkhaüser, Boston (1993). Zbl0816.49001MR1201152
  20. [20] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell’area. Rend. Mat. Appl.8 (1975) 277–294. Zbl0316.35036MR375037
  21. [21] E. De Giorgi, Γ-convergenza e G-convergenza. Boll. Un. Mat. Ital. 14-A (1977) 213–220. Zbl0389.49008
  22. [22] N. Dunford and J.T. Schwartz, Linear operators. Part I. General theory. Wiley-Interscience Publication, New York (1988). Zbl0084.10402MR1009162
  23. [23] 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). Zbl0455.35010
  24. [24] 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. Zbl0737.73009MR1145750
  25. [25] E.Ya. Khruslov and V.A. Marchenko, Homogenization of Partial Differential Equations, vol. 46. Progr. Math. Phys. Birkhäuser, Boston (2006). Zbl1113.35003MR2182441
  26. [26] N.G. Meyers, 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. Zbl0127.31904MR159110
  27. [27] U. Mosco, Composite media and asymptotic Dirichlet forms. J. Func. Anal.123 (1994) 368–421. Zbl0808.46042MR1283033
  28. [28] 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. Zbl0920.35019
  29. [29] C. Pideri and P. Seppecher, A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn.9 (1997) 241–257. Zbl0893.73006MR1482641
  30. [30] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa Cl. Sci.22 (1968) 571–597. Zbl0174.42101MR240443
  31. [31] L. Tartar, The General Theory of Homogenization: A Personalized Introduction. Lect. Notes Unione Matematica Italiana. Springer-Verlag, Berlin, Heidelberg (2009). Zbl1188.35004MR2582099

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