A Remark on the Stability of the Determinant in Bidimensional Homogenization

Fernando Farroni; François Murat

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 1, page 209-215
  • ISSN: 0392-4041

Abstract

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For conductivity problems in dimension N = 2, we prove a variant of a classical result: if a sequence A ϵ of matrices H-converges to A 0 (or in other terms if A ϵ converges to A 0 in the sense of homogenization) and if d ⁢ e ⁢ t ⁢ A ϵ tends to c 0 a.e., then one has d ⁢ e ⁢ t ⁢ A 0 = c 0 .

How to cite

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Farroni, Fernando, and Murat, François. "A Remark on the Stability of the Determinant in Bidimensional Homogenization." Bollettino dell'Unione Matematica Italiana 3.1 (2010): 209-215. <http://eudml.org/doc/290687>.

@article{Farroni2010,
abstract = {For conductivity problems in dimension N = 2, we prove a variant of a classical result: if a sequence $A^\{\epsilon\}$ of matrices H-converges to $A^\{0\}$ (or in other terms if $A^\{\epsilon\}$ converges to $A^\{0\}$ in the sense of homogenization) and if $det \, A^\{\epsilon\}$ tends to $c^\{0\}$ a.e., then one has $det \, A^\{0\} = c^\{0\}$.},
author = {Farroni, Fernando, Murat, François},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {209-215},
publisher = {Unione Matematica Italiana},
title = {A Remark on the Stability of the Determinant in Bidimensional Homogenization},
url = {http://eudml.org/doc/290687},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Farroni, Fernando
AU - Murat, François
TI - A Remark on the Stability of the Determinant in Bidimensional Homogenization
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/2//
PB - Unione Matematica Italiana
VL - 3
IS - 1
SP - 209
EP - 215
AB - For conductivity problems in dimension N = 2, we prove a variant of a classical result: if a sequence $A^{\epsilon}$ of matrices H-converges to $A^{0}$ (or in other terms if $A^{\epsilon}$ converges to $A^{0}$ in the sense of homogenization) and if $det \, A^{\epsilon}$ tends to $c^{0}$ a.e., then one has $det \, A^{0} = c^{0}$.
LA - eng
UR - http://eudml.org/doc/290687
ER -

References

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  2. BRIANE, M. - MANCEAU, D. - MILTON, G. W., Homogenization of the two-dimensional Hall effect, J. Math. Anal. Appl., 339 , 2 (2008), 1468-1484. Zbl1206.78090MR2377101DOI10.1016/j.jmaa.2007.07.044
  3. DYKHNE, A. M., Conductivity of a two dimensional two-phase system, A. Nauk. SSSR, 59 (1970), 110-115. 
  4. FRANCFORT, G. A. - MURAT, F., Optimal bounds for conduction in two-dimensional, two-phase, anisotropic media, in Nonclassical continuum mechanics (Durham, 1986), R. J. Knops and A. A. Lacey, eds., London Mathematical Society Lecture Notes Series122 (Cambridge University Press, Cambridge, 1987), 197-212. MR926503DOI10.1017/CBO9780511662911.013
  5. KELLER, J. B., A theorem on conductivity of composite medium, J. Mathematical Phys., 5, 4 (1964), 548-549. Zbl0129.44001MR161559DOI10.1063/1.1704146
  6. MARINO, A. - SPAGNOLO, S., Un tipo di approssimazione dell'operatore ∑ i , j = 1 n D i ⁢ ( a i ⁢ j ⁢ ( x ) ⁢ D j ) con operatori ∑ j = 1 n D j ⁢ ( β ⁢ ( x ) ⁢ D j ) , Annali della Scuola Normale Superiore di Pisa23, 4 (1969), 657-673. MR278128
  7. MENDELSON, K. S., A theorem on the effective conductivity of a two-dimensional heterogeneous medium, J. of Applied Physics, 46 , 11 (1975), 4740-4741. 
  8. MILTON, G. W., The theory of composites, Cambridge Monographs in Applied and Computational Mathematics6 , Cambridge University Press (Cambridge, 2002). Zbl0993.74002MR1899805DOI10.1017/CBO9780511613357
  9. MURAT, F., H-convergence, Séminaire d'analyse fonctionnelle et numérique 1977-78, Université d'Alger. English translation: MURAT, F. - TARTAR, L., H-convergence, in Topics in the mathematical modelling of composite materials, A. Cherkaev - R. V. Kohn, eds., Progress in Nonlinear Differential Equations and their Applications, 31 (Birkhäuser, Boston, 1997), 21-43. MR1493039
  10. SBORDONE, C., Γ -convergence and G-convergence, Rend. Sem. Mat. Univ. Politec. Torino, 40 (1983), 25-51. MR724198
  11. SPAGNOLO, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Annali della Scuola Normale Superiore di Pisa, 22, 3 (1968), 571-597. MR240443
  12. TARTAR, L., The general theory of homogenization, A personalized introduction, Lecture Notes of the Unione Matematica Italiana, 7 (Springer-Verlag, Berlin, 2010). Zbl1188.35004MR2582099DOI10.1007/978-3-642-05195-1

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