Homogenization of the Maxwell Equations: Case I. Linear Theory

Abstract

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Homogenization of the Maxwell Equations: Case II. Nonlinear Conductivity

“…In this nonlinear case we need different function spaces for the different quantities in the Maxwell equations, which we did not need in the linear case in [17]. This is clearly seen in the a priori estimates in Section 6 and in the following observation.…”

“…By C we denote any fixed constant which may take different values in any place it appears in an equation or inequality. Definitions of function spaces used in this paper are found in [17] and [19]. We define the Y -cell as the open unit cube…”

“…Homogenization results for linear Maxwell's equations can be found in [2], [3], [10], [12], [14] and [20]. In [17] the two-scale convergence method was used which turned out to give simple and straightforward proofs in the homogenization procedure.…”

The Subgrid Modeling for Maxwell's Equations With Multiscale Isotropic Random Conductivity and Permittivity

“…Methods of homogenization are often applied to Maxwell's equations. In these methods it is assumed that the medium is ε-periodic in the sense that it can be viewed as the union of a collection of disjoint open identical cubes with side length ε [4]. Such assumptions are correct for composite materials.…”

Frequency Dispersion of Dielectric Permittivity and Electric Conductivity of Rocks via Two-Scale Homogenization of the Maxwell Equations

“…It was proved in many studies that the macroscopic Maxwell equations can be strongly different from the microscopic ones: instantaneous material laws turn into constitutive laws with memory [13,[17][18][19][20][21][22][23][24][25]. More general case has been considered in [26], with polarization of composite ingredients being not instantaneous but obeying the Debye or Lorenz polarization laws with relaxation.…”