Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions

Magnetostatic and electrostatic problems with mixed boundary conditions are studied. The medium can have a nonsmooth boundary and very irregular physical properties due to inhomogeneity and anisotropy. The topological assumptions are general enough to meet the requirements of the engineering applications. Necessary and sufficient conditions for solvability are found and the set of the solutions is characterized. Moreover, uniqueness is recovered by means of a finite number of supplementary conditions which are equivalent to prescribing a finite number of suitably chosen fluxes or potentials. A functional framework in which other important problems of electromagnetics naturally fit is developed.