Well-conditioned boundary integral formulations for the ... - Njit

Taking into account the fact that |ξ · â(ξ)| 2 ≤ |ξ| 2 |â(ξ)| 2 , the positivity of the expression in the
left-hand side of equations (38) follows immediately once we establish that
κ 2 R{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 } + κ 1 I{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 }
(
≥ |ξ| 2 κ1
κ 2 1 + I{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 } − κ )
2
κ2 2
κ 2 1 + R{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 } . (39)
κ2 2
The inequality (39) can be seen to be equivalent to the inequality
(
I (κ 1 + iκ 2 )[|ξ| 2 − (κ 1 + iκ 2 ) 2 ] −1/2) ( )
|ξ|
2
≥ I [|ξ| 2 − (κ 1 + iκ 2 ) 2 ] −1/2
κ 1 + iκ 2
which in turn is equivalent to the inequality
( )
−κ1 + iκ 2
I
κ 2 1 + [|ξ| 2 − (κ 1 + iκ 2 ) 2 ] 1/2 } ≥ 0. (40)
κ2 2
Given that R{[|ξ| 2 − (κ 1 + iκ 2 ) 2 ] 1/2 } > 0 and I{[|ξ| 2 − (κ 1 + iκ 2 ) 2 ] 1/2 } < 0, the inequality (40)
follows immediately, and the proof is complete.
2.1 Principal symbol regularizing operators
We present in what follows another class of regularizing operators that consists of operators that
have the same principal symbol in the sense of pseudodifferential operators [48, 54] as the operators
R defined in equations (11). The calculation of the principal symbols of the operators R defined
in equations (11) is based on the principal symbols of scalar and vector single layer operators S K
and π Γ S K for complex wavenumbers K such that IK > 0, where the projection operator onto the
tangent space of Γ is defined by π Γ = (n×·)×n. The principal symbols of these operators are equal
1
to √ and √ 1
I 2 |ξ| 2 −K 2 2 |ξ| 2 −K 2 2 respectively [4, 17, 19, 49], where we view the operator π Γ S k in terms
of the Helmholtz decomposition and I 2 stands for the identity 2 × 2 matrix—we assume without
loss of generality that Γ is simply connected. In the previous equations the variable ξ ∈ T M ∗ (Γ)
represents the Fourier symbol of the tangential gradient operator ∇ Γ , and T M ∗ (Γ) represents the
cotangent bundle of Γ [55]. Given that the principal symbol of the operators 1 2 (−∆ Γ − K 2 ) − 1 2
is equal to
1
2 √ |ξ| 2 −K 2 and the principal symbol of the operator 1 2 (−∆ Γ − K 2 ) − 1 2 I 2 is equal to
1
2 √ |ξ| 2 −K 2 I 2 [55], the previous statement can be expressed more precisely in the form [4, 17, 49]
S K = 1 2 (−∆ Γ − K 2 ) − 1 2 mod Ψ −3 (Γ), π Γ S K = 1 2 (−∆ Γ − K 2 ) − 1 2 I2 mod Ψ −3 (TM(Γ)) (41)
where Ψ −3 (Γ) denotes the operator algebra of pseudodifferential operators of order −3 on Γ and
Ψ −3 (T M(Γ)) denotes the operator algebra of pseudodifferential operators of order −3 on the space
of tangential vector fields on T M(Γ). The meaning of the notation A = B mod Ψ s (Γ) where
A and B are pseudodifferential operators defined on scalar functions on Γ is that A − B is a
pseudodifferential operator of order s, that is A − B : H p (Γ) → H p+s (Γ); the definition is similar
in the case when the operators A and B are tangential pseudodifferential operators. It follows
immediately then from equations (41) that
( ) T (
) (
n×S K = 1 ∇Γ
0 (−∆
−−→
Γ − K 2 ) − 1 2 ∆
−1
2 curl Γ −(−∆ Γ − K 2 ) − Γ
div )
Γ
1
2 0
−∆ −1
Γ
curl mod Ψ −3 (TM(Γ)).
Γ
(42)
14