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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.

MSC:
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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