An initial-boundary value problem for the uniform system of Maxwell's equations in the case of a magnetodielectric body with conductive ferromagnetic inclusions

The uniform system of electrodynamics equations solved for strength derivatives with respect to time is considered as applied to the case of a heterogeneous magnetodielectric with foreign metallic ferromagnetic inclusions. It is assumed that the magnetodielectric and ferromagnetic inclusions have a piecewise smooth boundaries, and the closed domains occupied by the ferromagnetics do not intersect and are included in the domain occupied by the magnetodielectric. The electromagnetic characteristics of individual media satisfy the natural requirements of continuity. Under these assumptions, the differential operator $\hat{A}$ defining the right part of the system of Maxwell's equations, is explored. For the operator $\hat{A}$ we selected the most natural definition domain: the space of ordered pairs of vector fields square summable together with their generalized curls. It is shown that such a choice of the definition domain of operator $\hat{A}$ takes into account the boundary conditions of continuity of tangent components of the intensities. It is proved that the operator $\hat{A}$ is closed and has an important spectral property: operator $(\hat{A}-p\hat{I})^{-1}$ ($\hat{I}$ is the identity operator) is defined on the space of ordered pairs of square summable vector fields and his norm is smaller or equal to $1/p$. Based on the Hille–Yosida theorem, we conclude that the studied initial-boundary value problem has a unique solution if differentiability with respect to time is meant as differentiability with respect to the mean-square norm.