ON MAXWELL EQUATIONS WITH THE TRANSPARENT ...

Maxwell Equations with the Transparent Boundary Condition 291
By the Lax-Milgram lemma we know that the problem (2.11) has a unique solution. To show
the stability estimate (2.15), we know from (2.11) that
Therefore, by (2.16),
|a(E,E)| ≤ |k| −2 ‖J s ‖ L 2 (Ω R)‖ kE ‖ L 2 (Ω R).
‖ ∇ × E ‖ L 2 (Ω R) + ‖ kE ‖ L 2 (Ω R) ≤ Ck −1
2 ‖J s ‖ L 2 (Ω R).
This completes the proof.
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3. The Maxwell Scattering Problem
We first give the assumptions required on the boundary and initial data:
(H1) E 0 ,H 0 , ∇ × E 0 , ∇ × H 0 ∈ H(curl; Ω R ) and supp(E 0 ), supp(H 0 ) ⊂ B R ;
(H2) J ∈ H 1 (0, T; L 2 (Ω R )) 3 , J| t=0 = 0, and supp(J) ⊂ B R × (0, T).
In the rest of this paper, we will always assume that J is extended so that
J ∈ H 1 (0, +∞; L 2 (Ω R )) 3 ,
‖J‖ H1 (0,+∞;L 2 (Ω R)) ≤ C‖J‖ H1 (0,T;L 2 (Ω R)).
The following lemma can be proved by the standard energy argument.
Lemma 3.1. Let Ê,Ĥ be the solution of the following problem
Then
ε ∂Ê
∂t − ∇ × Ĥ = 0
in Ω R × (0, T),
µ ∂Ĥ
∂t + ∇ × Ê = 0 in Ω R × (0, T),
n × Ê = 0 on Γ D ∪ Γ R ,
Ê| t=0 = E 0 , Ĥ| t=0 = H 0 .
‖Ê‖ L 2 (Ω R) + ‖Ĥ‖ L 2 (Ω R) ≤ C‖E 0 ‖ L 2 (Ω R) + C‖H 0 ‖ L 2 (Ω R),
‖ ∂ t Ê ‖ L2 (Ω R) + ‖ ∂ t Ĥ ‖ L2 (Ω R) ≤ C‖ ∇ × E 0 ‖ L2 (Ω R) + C‖ ∇ × H 0 ‖ L2 (Ω R),
‖ ∂ 2 tÊ ‖ L 2 (Ω R) + ‖ ∂ 2 t Ĥ ‖ L 2 (Ω R) ≤ C‖ ∇ × ∇ × E 0 ‖ L 2 (Ω R) + C‖ ∇ × ∇ × H 0 ‖ L 2 (Ω R).
Let E ′ = E − Ê,H′ = H − Ĥ. Then by (1.1)-(1.2) we know that
ε ∂E′
∂t − ∇ × H′ = J in [R 3 \ ¯D] × (0, T), (3.1)
µ ∂H′
∂t
The boundary condition (1.3) becomes
+ ∇ × E ′ = 0 in [R 3 \ ¯D] × (0, T). (3.2)
n × E ′ = 0 on Γ D × (0, T). (3.3)