Weak and strong formulations for the time-harmonic eddy-current problem in general domains

The eddy-current problem for the time-harmonic Maxwell equations in domains and with conductors of general topology is considered. The existence of a unique magnetic field is proved for a suitable weak formulation. An equivalent strong formulation is then derived, where the conditions related to the specific geometry of the domain are made explicit. In particular, a new condition that must be satisfied by the magnetic field on the interface between a multiply-connected conductor and the non-conducting region is determined. Finally, the strong formulation of the problem for the electric field in the non-conducting region is derived, and the existence of a unique solution is proved. In conclusion, this leads to the determination of the complete set of equations describing the eddy-current problem in terms of the magnetic and the electric fields. Whether some commonly-used formulations satisfy the additional condition on the interface is also checked.