ON MAXWELL EQUATIONS WITH THE TRANSPARENT ...

Maxwell Equations with the Transparent Boundary Condition 295
Denote ã mn = a mn χ [0,T] ,˜b mn = b mn χ [0,T] , where χ [0,T] is the characteristic function of the
interval (0, T). Therefore
∫ T
e −2s1t 〈(L −1 ◦ G e ◦ L )(x × E), ˆx × E × ˆx〉 ΓR dt
0
∑
∞ n∑
∫ [ (
∞
√εµsRh (1)
= R 2 e −2s1t L −1 n (i √ ) ]
εµsR)
n=1 m=−n −∞
z n (1) (i √ ∗ ˜b mn
¯˜bmn
εµsR)
∫ [ (
∞
+ e −2s1t L −1 z n (1) (i √ ) ]
εµsR)
√ (1)
−∞
εµsRh n (i √ ∗ ã mn
¯ã mn .
εµsR)
Note that by the formula for the inverse Laplace transform we have
g(t) = F −1 (e s1t L (g)(s 1 + is 2 )),
where F −1 denotes the inverse Fourier transform with respect to s 2 . By the Plancherel identity
we then obtain
∫ T
e −2s1t 〈(L −1 ◦ G e ◦ L )(x × E), ˆx × E × ˆx〉 ΓR dt
0
∑
∞ n∑
∫ (
∞ √εµsRh (1)
= 2πR 2 n (i √ )
εµsR)
n=1 m=−n −∞ z n (1) (i √ |L (˜b mn )| 2
εµsR)
∫ (
∞
z n (1) (i √ )
εµsR)
+ √ (1)
−∞ εµsRh n (i √ |L (ã mn )| 2 .
εµsR)
Since k = i √ εµs satisfies Im(k) > 0, by using Lemmas 2.3 and 2.4 we obtain
−Re
∫ T
0
e −2s1t 〈(L −1 ◦ G e ◦ L )(x × E), ˆx × E × ˆx〉 ΓR dt ≥ 0.
For any 0 < t ∗ < T, by taking Φ = e −2s1t Eχ (0,t ∗ ) in (3.9), Ψ = e −2s1t Hχ (0,t ∗ ) in (3.10), and
adding the two equations, we obtain
1
2
∫ t
∗
0
e −2s1t d dt
(
)
ε‖E‖ 2 L 2 (Ω + R) µ‖H‖2 L 2 (Ω R) dt ≤
∫ t
∗
0
e −2s1t (J,E)dt.
By standard argument we can deduce
[ ( )]
max e −2s1t ε‖E‖ 2 L
0≤t≤T
2 (Ω + R) µ‖H‖2 L 2 (Ω R)
(
)
≤ C ε‖E 0 ‖ 2 L 2 (Ω + µ‖H R) 0 ‖ 2 L 2 (Ω R) + C‖ e −s1t J ‖ L 1 (0,T;L 2 (Ω R)).
By letting s 1 → 0, we obtain
(
)
max ε‖E‖ 2 L
0≤t≤T
2 (Ω + R) µ‖H‖2 L 2 (Ω R)
(
)
≤ C ε‖E 0 ‖ 2 L 2 (Ω + µ‖H R) 0 ‖ 2 L 2 (Ω R) + C‖J‖ L1 (0,T;L 2 (Ω R)).
(3.18)
Since E 0 ,H 0 has a compact support inside B R , a mn (R, 0) = b mn (R, 0) = 0 on Γ R . By differentiating
(1.8) in time we know that
√ µ
ε ˆx × ∂ tH = (L −1 ◦ G e ◦ L )(ˆx × ∂ t E| ΓR ) on Γ R .