Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient

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.