Abstract
The notion of a Hall matrix associated with a possibly anisotropic conducting material in the presence of a small magnetic field is introduced. 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 Murat–Tartar. Extending a result of Bergman, 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.
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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)
Bauman P., Marini A., Nesi V.: Univalent solution of an elliptic system of partial differential equations arising in homogenization. Indiana Univ. Math. J. 50(2), 747–757 (2001)
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
Briane M., Manceau D., Milton G.W.: Homogenization of the two-dimensional Hall effect. J. Math. Ana. App. 339, 1468–1484 (2008)
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)
Colombini F., Spagnolo S.: Sur la convergence de solutions d’équations paraboliques. J. Math. Pures et Appl. 56, 263–306 (1977)
Dacorogna B.: Direct Methods in the Calculus of Variations, in Applied Mathematical Sciences 78. Springer, Berlin (1989)
Lakes R.: Cellular solid structures with unbounded thermal expansion. J. Mater. Sci. Lett. 15, 475–477 (1996)
Landau L., Lifchitz E.: Électrodynamique des Milieux Continus. Éditions Mir, Moscou (1969)
Meyers N.G.: An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17, 189–206 (1963)
Murat F., Tartar L.: H-convergence, Topics in the Mathematical Modelling of Composite Materials. In: Cherkaev, L., Kohn, R.V.(eds) Progress in Nonlinear Differential Equations and their Applications, pp. 21–43. Birkaüser, Boston (1998)
Ali Omar, M.: Elementary Solid State Physics. Addison Wesley, Reading, MA, World Student Series Edition, 1975
Sigmund O., Torquato S.: Composites with extreme thermal expansion coefficients. Appl. Phys. Lett. 69, 3203–3205 (1996)
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)
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)
Levi–Civita symbol, Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Levi-Civita_symbol
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Communicated by G. Dal Maso
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Briane, M., Milton, G.W. Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient. Arch Rational Mech Anal 193, 715–736 (2009). https://doi.org/10.1007/s00205-008-0200-y
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DOI: https://doi.org/10.1007/s00205-008-0200-y