Electromagnetic Processes in Doubly-Nonlinear Composites

Abstract

Electromagnetic processes in inhomogeneous conductors are here described by coupling the Maxwell equations with nonlinear constitutive relations of the form and , neglecting hysteresis and displacement currents. The latter equality may also account for the Hall effect.

A doubly-nonlinear parabolic-hyperbolic equation is formulated, and existence of a solution is proved via approximation by time-discretization, derivation of a priori estimates, and passage to the limit via compensated compactness and compactness by strict convexity.

It is then assumed that the medium is a composite that exhibits periodic oscillations in space. Convergence to a corresponding homogenized two-scale problem is proved as the oscillation period vanishes, via Nguetseng's notion of two-scale convergence. Finally this two-scale formulation is proved to be equivalent to a coarse-scale problem.

Keywords:

• Electromagnetism
• Homogenization
• Maxwell equations
• Two-scale convergence

Mathematics Subject Classification:

• 35K60
• 35R35
• 78A25
• 78M40

Acknowledgment

This research was partially supported by the project “Free boundary problems, phase transitions and models of hysteresis” of Italian M.U.R..

Notes

¹Some authors write in place of in (2.10), but then should be assumed to be proportional to .

²It is known that for r = ∞ difficulties arise in defining these spaces, for L ^∞(Ω) and L ^∞(Ω × 𝒴) are not separable. This drawback may be removed by assuming that these vector-valued functions are weakly star measurable.