Maxwell’s equations in inhomogeneous bi-anisotropic materials: Existence, uniqueness and stability for the initial value problem

In the present paper inhomogeneous bi-anisotropic materials characterized by matrices of electric permittivity, magnetic permeability and magnetoelectric characteristics are considered. All elements of these matrices are functions of the position in three dimensional space. The time-dependent Maxwell’s equations describe the electromagnetic wave propagation in these materials. Maxwell’s equations together with zero initial data are analyzed and a statement of the initial value problem (IVP) is formulated. This IVP is reduced to the IVP for a symmetric hyperbolic system of partial differential equations of the first order. Applying the theory of a symmetric hyperbolic system, new existence, uniqueness and stability estimate theorems have been obtained for the IVP of Maxwell’s equations in inhomogeneous bi-anisotropic materials.