Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient

Marc Briane (1) , Graeme W. Milton (2, 3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Department of Mathematics - University of Utah
3 Department of Mathematics

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, it is shown 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 a.e., while the homogenized Hall matrix is negative isotropic.

Mots clés: Homogenization; Hall effect

Domaines: Equations aux dérivées partielles [math.AP]

Marc Briane, Graeme W. Milton. Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient. Archive for Rational Mechanics and Analysis, 2009, 193 (3), pp.715-738. ⟨10.1007/s00205-008-0200-y⟩. ⟨hal-00130591v2⟩