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Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient. (English) Zbl 1170.74019
Summary: We introduce the notion of a Hall matrix associated with a possibly anisotropic conducting material in the presence of a small magnetic field. Then, for any material having a microstructure, we prove a general homogenization result satisfied by the Hall matrix in the framework of the \(H\)-convergence of F. Murat and L. Tartar [in: L. Cherkaev and R. V. Kohn (eds.), Progress in nonlinear differential equations and their applications. Boston: Birkhäuser. 21–43 (1998)]. Extending a result of D. J. Bergman [in G. Deutscher et al. (eds.), Percolation structures and processes. 297–321 (1983)], we show that the Hall matrix can be computed from the corrector associated with the homogenization problem when no magnetic field is present. Finally, we give an example of a microstructure for which the Hall matrix is positive isotropic almost everywhere, while the homogenized Hall matrix is negative isotropic.

74F15 Electromagnetic effects in solid mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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[1] Ancona, A., Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators, Nagoya Math. J., 165, 123-158, (2002) · Zbl 1028.31003
[2] Bauman, P.; Marini, A.; Nesi, V., Univalent solution of an elliptic system of partial differential equations arising in homogenization, Indiana Univ. Math. J., 50, 747-757, (2001) · Zbl 1330.35121
[3] Bergman, D.J.: Self duality and the low field Hall effect in 2D and 3D metal-insulator composites. In: Deutscher, G., Zallen, R., Adler, J. (eds.) Percolation Structures and Processes, pp. 297-321, 1983
[4] Briane, M.; Manceau, D.; Milton, G. W., Homogenization of the two-dimensional Hall effect, J. Math. Ana. App., 339, 1468-1484, (2008) · Zbl 1206.78090
[5] Briane, M.; Milton, G. W.; Nesi, V., Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity, Arch. Rational Mech. Anal., 173, 133-150, (2004) · Zbl 1118.78009
[6] Colombini, F.; Spagnolo, S., Sur la convergence de solutions d’équations paraboliques, J. Math. Pures et Appl., 56, 263-306, (1977) · Zbl 0354.35009
[7] Dacorogna B.: Direct Methods in the Calculus of Variations, in Applied Mathematical Sciences 78. Springer, Berlin (1989) · Zbl 0703.49001
[8] Lakes, R., Cellular solid structures with unbounded thermal expansion, J. Mater. Sci. Lett., 15, 475-477, (1996)
[9] Landau L., Lifchitz E.: Électrodynamique des Milieux Continus. Éditions Mir, Moscou (1969)
[10] Meyers, N. G., An \(L p\)-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Sup. Pisa, 17, 189-206, (1963) · Zbl 0127.31904
[11] Murat, F.; Tartar, L.; Cherkaev, L. (ed.); Kohn, R. V. (ed.)\(, H\)-convergence, Topics in the Mathematical Modelling of Composite Materials, 21-43, (1998), Boston
[12] Ali Omar, M.: Elementary Solid State Physics. Addison Wesley, Reading, MA, World Student Series Edition, 1975
[13] Sigmund, O.; Torquato, S., Composites with extreme thermal expansion coefficients, Appl. Phys. Lett., 69, 3203-3205, (1996)
[14] Sigmund, O.; Torquato, S., Design of materials with extreme thermal expansion using a three-phase topology optimization method, J. Mech. Phys. Solids, 45, 1037-1067, (1997)
[15] Stroud, D.; Bergman, D. J., New exact results for the Hall-coefficient and magnetoresistance of inhomogeneous two-dimensional metals, Phys. Rev. B (Solid State), 30, 447-449, (1984)
[16] Levi-Civita symbol, Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Levi-Civita_symbol
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