Geodesic
A Remark on the Stability of the Determinant in Bidimensional Homogenization
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 1, pp. 209-215.

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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 $det \, A^{0} = c^{0}$. 
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Farroni, Fernando; Murat, François. A Remark on the Stability of the Determinant in Bidimensional Homogenization. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 1, pp. 209-215. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_1_a9/

[1] L. Boccardo - F. Murat, Homogénéisation de problèmes quasi-linéaires, in Atti del convegno: Studio di problemi-limite della analisi funzionale, Bressanone, settembre 1981, Pitagora, Bologna (1982), 13-51.

[2] M. Briane - D. Manceau - G. W. Milton, Homogenization of the two-dimensional Hall effect, J. Math. Anal. Appl., 339 , 2 (2008), 1468-1484. | DOI | MR | Zbl

[3] A. M. Dykhne, Conductivity of a two dimensional two-phase system, A. Nauk. SSSR, 59 (1970), 110-115.

[4] G. A. Francfort - F. Murat, 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 Series 122 (Cambridge University Press, Cambridge, 1987), 197-212. | DOI | MR

[5] J. B. Keller, A theorem on conductivity of composite medium, J. Mathematical Phys., 5, 4 (1964), 548-549. | DOI | MR | Zbl

[6] A. Marino - S. Spagnolo, Un tipo di approssimazione dell'operatore $\sum_{i,j = 1}^{n} \, D_{i}(a_{ij}(x) \, D_{j})$ con operatori $\sum_{j = 1}^{n} \, D_{j}(\beta(x) \, D_{j})$, Annali della Scuola Normale Superiore di Pisa 23, 4 (1969), 657-673. | fulltext EuDML | MR

[7] K. S. Mendelson, A theorem on the effective conductivity of a two-dimensional heterogeneous medium, J. of Applied Physics, 46 , 11 (1975), 4740-4741.

[8] G. W. Milton, The theory of composites, Cambridge Monographs in Applied and Computational Mathematics 6 , Cambridge University Press (Cambridge, 2002). | DOI | MR | Zbl

[9] F. Murat, H-convergence, Séminaire d'analyse fonctionnelle et numérique 1977-78, Université d'Alger. English translation: F. Murat - L. Tartar, 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. | MR

[10] C. Sbordone, $\Gamma$-convergence and G-convergence, Rend. Sem. Mat. Univ. Politec. Torino, 40 (1983), 25-51. | MR

[11] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Annali della Scuola Normale Superiore di Pisa, 22, 3 (1968), 571-597. | fulltext EuDML | MR

[12] L. Tartar, The general theory of homogenization, A personalized introduction, Lecture Notes of the Unione Matematica Italiana, 7 (Springer-Verlag, Berlin, 2010). | DOI | MR | Zbl