Electromagnetics in deterministic and stochastic bianisotropic media

Theorem
Homogenisation (deterministic media)
Assume that the family of matrices A(y, p) satisfies
for A = A L, A −1
L
(Re A(y, p))u · u ≥ c�u� 2 , y ∈ Y , p ∈ C+, u ∈ R 6 . (42)
(defined in (41)).
The solution u ɛ = (E ɛ , H ɛ ) tr of (A ɛ oru ɛ + G ɛ d ⋆ u ɛ ) ′ = Mu ɛ + j with zero initial
conditions and the perfect conductor boundary condition satisfies
u ɛ ∗ ⇀ u ∗ , in L ∞ ([0, T ], X),
where u ∗ = (E ∗ , H ∗ ) tr is the unique solution of the homogeneous Maxwell
system
(d ∗ ) ′ = Mu ∗ + j, in (0, T ] × O, (43)
with zero initial conditions and the perfect conductor boundary condition, and
subject to the constitutive relations
such that A h or + � G h d = A h L, where
d ∗ = A h oru ∗ + G h d ⋆ u ∗
A h �
εh L
L =
ζ h L
ξh L
µ h �
,
L
I. G. Stratis (Maths Dept, NKUA) Collège de France November 18, 2011 50 / 95
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