c Pleiades Publishing, Ltd., 2011. ISSN 0012-2661, Differential Equations, 2011, Vol. 47, No. 9, pp. 1297–1306. c A.V. Baryshev, Yu.A. Eremin, 2011, published in Differentsial’nye Uravneniya, 2011, Vol. 47, No. 9, pp. 1284–1293. Original Russian Text
INTEGRAL EQUATIONS
Justification of Integral Representations for Fields in a Diffraction Problem on a Cluster of Particles in a Plane-Layered Medium A. V. Baryshev and Yu. A. Eremin Moscow State University, Moscow, Russia Received April 13, 2011
Abstract— We consider a problem on the diffraction of electromagnetic field on a cluster of permeable particles placed in a plane-layered medium. The completeness and closedness of the system of basis functions used in the representation of the fields is proved, and the convergence of the approximate solution to the exact solution of the original boundary value diffraction problem is shown. DOI: 10.1134/S0012266111090072
1. INTRODUCTION Problems of electromagnetic wave diffraction by a system of several local inhomogeneities lying entirely in one layer of a plane-layered medium occur in various fields of natural sciences. The recent interest in such problems is related to studies in the field of subsurface radars [1] and the analysis of scattering characteristics of optic antennas [2]. The main tool in the analysis of such problems is mathematical modeling based on the solution of boundary value problems for the system of Maxwell equations. An efficient numerical solution of these diffraction problems can be constructed on the basis of integral [3] and integro-functional methods like the null field method and the discrete source method [4]. A mathematical model for the analysis of scattering properties of a cluster of permeable particles placed in a plane-layered medium was developed and implemented in [5]. In the present paper, we carry out a mathematical justification of integral field representations constructed on the basis of the discrete source method. We prove the completeness and closedness of the corresponding system of basis functions and the convergence of the approximate solution to the exact solution of the original boundary value electromagnetic wave diffraction problem. 2. STATEMENT OF THE BOUNDARY VALUE DIFFRACTION PROBLEM For brevity, we consider a cluster consisting of two scatterers. The case of a large number of inhomogeneities can be considered in a similar way. Assume that the plane-layered medium consists of three domains, D1 (z < 0), Df (0 < z < d), and D0 (d < z); the domains D1 and Df are separated by a plane Ξ1 , and the domains D1 and Df are separated by a plane Ξ0 . We introduce a Cartesian coordinate system such that the origin lies on the boundary between the domains D1 and Df and the axis OZ is directed towards the domain D0 and is perpendicular to the boundaries Ξ1 and Ξ0 . We assume that there are two scatterers, whose interiors will be denoted by Di1 and Di2 , having smooth boundary ∂Di1 ,i2 ∈ C 2,α and lying entirely in the domain Df . For the external excitation we take a plane electromagnetic wave {E0 , H0 } propagating from the domain D1 at an angle θ1 to the axis OZ. By solving the problem on the refraction and reflection of the plane wave field {E0 , H0 } on the interface Ξ1,0 , we find the external excitation field {E0f , H0f } inside the film; a closed-form expression for this field can be found, e.g., in the monograph [6, p. 207]. Let 1297
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{E1,f,0 , H1,f,0 } be the scattered field in the corresponding domain D1,f,0 , and let {Ei1 ,i2 , Hi1 ,i2 } be the complete field inside the scatterers Di1 ,i2 . Then the mathematical statement of the diffraction problem is a boundary value problem for the system of Maxwell equations and has the form rot Hζ = jkεζ Eζ , rot Eζ = −jkμζ Hζ in Dζ , ζ = 0, f, 1, i1 , i2 , nP × (Ei1 (P ) − Ef (P )) = nP × E0f (P ), nP × (Hi1 (P ) − Hf (P )) = nP × H0f (P ), P ∈ ∂Di1 , (1) nP × (Ei2 (P ) − Ef (P )) = nP × E0f (P ), nP × (Hi2 (P ) − Hf (P )) = nP × H0f (P ), ez × (E0 (P ) − Ef (P )) = 0, ez × (H0 (P ) − Hf (P )) = 0, P ∈ Ξ0 , ez × (Ef (P ) − E1 (P )) = 0, ez × (Hf (P ) − H1 (P )) = 0, P ∈ Ξ1 , √ r √ εζ Eζ × − μζ Hζ = 0, r = |M | → ∞, ζ = 0, 1, lim r→∞ r (|Ef |, |Hf |) = o(exp{−| Im (kf )|}), = x2 + y 2 → ∞.
P ∈ ∂Di2 ,
Here ez is the unit z-vector of the Cartesian coordinate system, and nP is the normal to the surface ∂Di1 ,i2 at the point P . The dependence of the fields on time is chosen in the form ejωt . In addition, we assume that the parameters of the media satisfy the condition Im (kζ2 ) ≤ 0. In this case, the boundary value problem has the unique solution (see [3]). 3. DISCRETE SOURCE METHOD An approximate solution of problem (1) will be constructed on the basis of the fundamental concept of the discrete source method. The idea of the method is the following: the scattered field in the exterior of inhomogeneities and the complete field inside them are constructed in the form of a finite linear combination of fields of electric dipoles analytically satisfying the system of Maxwell equations in the respective domains, the transmission conditions for the tangential components of the fields on the interface Ξ0,1 , and the conditions at infinity for the scattered field in D1,f,0 . As a result, the solution of the boundary value problem (1) is reduced to the problem of the approximation of the external excitation field by the fields of given dipoles. The unknown amplitudes of the dipoles are found only from the transmission conditions for the tangential field components on the scatterer surfaces ∂Di1 ,i2 . The representation for the scattered field is determined by the Green tensor of the layered medium, ⎞ ⎛ 0 0 G11 ⎟ ⎜ G=⎝ 0 0 ⎠, G11 ∂g\∂x ∂g\∂y G33 where each component of the tensor can be represented by a Sommerfeld integral. Moreover, the spectral functions occurring in the Sommerfeld integral provide the validity of the transmission conditions on the interface Ξ0,1 . Closed-form expressions for the spectral functions can be found from these transmission conditions as well. Expressions for the components of the Green tensor and the spectral functions can be found in [5]. For the qth scatterer, q = 1, 2, the dipoles representing the scattered field in the exterior domain are localized at points Mnq,e and are arranged everywhere densely on some normal auxiliary surface Sqe , {Mnq,e } = Sqe , q = 1, 2. The dipoles forming the field inside the scatterer are localized at the points Mnq,i and are dense on some normal auxiliary surface Sqi , {Mnq,i } = Sqi , q = 1, 2. The surfaces Sqe and Sqi lie strictly inside the domain Diq , q = 1, 2. We assume that there are N dipoles representing the field outside and inside the scatterer on each of the surfaces Sqe and Sqi , q = 1, 2. We assume that at each of the points Mnq,e and Mnq,i , q = 1, 2, there are three linearly independent dipoles oriented in accordance with the cylindrical coordinate system. Then the vector potentials of the qth scatterer corresponding to each such dipole can be represented in the form (M, Mnq,e ) = {G(M, Mnq,e )el }, Al(q,e) n
Al(q,i) (M, Mnq,i ) = {j0 (M, Mnq,i )el }, n DIFFERENTIAL EQUATIONS
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where j0 (M, Mnq,i ) = j0 (kiq RM,Mnq,i ) is the Bessel spherical function of zero order, q = 1, 2. The indices e and i correspond to the vector potential for the scattered and complete fields, respectively, inside the scatterer. Thus, we seek an approximate solution of the diffraction problem (1) in the form EN e
=
N 3
p1,e nl
rot rot{G(M, Mn1,e )el }
+
n=1 l=1
EN i1 =
N 3
N 3
2,e p2,e nl rot rot{G(M, Mn )el },
n=1 l=1 1,i p1,i nl rot rot{j0 (M, Mn )el },
EN i2 =
n=1 l=1
j rot EN HN e = e , kμe
N 3
2,i p2,i nl rot rot{j0 (M, Mn )el },
(2)
n=1 l=1
j HN rot EN i1 = i1 , kμi1
HN i2 =
j rot EN i2 , kμi2
q,i where el is the unit vector of the cylindrical coordinate system, pq,e nl and pnl are the discrete source amplitudes to be determined, and μe = {μ1 , μf , μ0 } depending on in which of the domains D1,f,0 outside the scatterer the point M lies. Let us prove the completeness of the function system used for the representation of the approximation solution (2).
Theorem 1. Let {e1 , h1 , e2 , h2 } ∈ Hτ = Lτ2 (∂Di1 ) × Lτ2 (∂Di1 ) × Lτ2 (∂Di2 ) × Lτ2 (∂Di2 ), e,i be normal surfaces, and let {Mnζ,e } = Sζe and {Mnζ,i } = Sζi , ζ = 1, 2. In addition, let let S1,2 ζ,i ∂Di1 ,i2 ∈ C 2,α . Then for each δ > 0, there exist a pair {Ne , Ni } and sequences {pζ,e nl } and {pnl }, ζ = 1, 2, such that N EN eζ iζ − Ee ≤ δ, ζ = 1, 2. + n, N ζ HN − H h iζ e Lτ (∂D ) 2
iζ
Proof. It suffices to prove the closedness of the function system used in the solution (2) in the space Hτ ; i.e., it suffices to show that the closure of the range of the matrix operator Tˆ coincides with Hτ , or ker Tˆ∗ = {0}, where Tˆ =
N 3 [T1nl , T2nl , T3nl , T4nl ] n=1 l=1
and the column vectors T1 , T2 , T3 , and T4 have the form ⎡ T1nl
T3nl
[n, rot rot{j0 (M, Mn1,i )el }]
⎢ ⎥ ⎢ jkεi [n, rot{j0 (M, M 1,i )el }] ⎥ n 1 ⎢ ⎥, =⎢ ⎥ 0 ⎣ ⎦ 0 ⎤ ⎡ − [n, rot rot{G(M, Mn1,e )el }] ⎥ ⎢ ⎢ −jkεe [n, rot{G(M, M 1,e )el }] ⎥ n ⎥, ⎢ =⎢ ⎥ 1,e )e }] −[n, rot rot{G(M, M ⎦ ⎣ l n 1,e −jkεe [n, rot{G(M, Mn )el }]
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⎤
⎡
⎤
0
T2nl
T4nl
2011
⎥ ⎢ ⎥ ⎢ 0 ⎥, ⎢ =⎢ ⎥ 2,i ⎣ [n, rot rot{j0 (M, Mn )el }] ⎦ jkεi2 [n, rot{j0 (M, Mn2,i )el }] ⎤ ⎡ − [n, rot rot{G(M, Mn2,e )el }] ⎥ ⎢ ⎢ −jkεe [n, rot{G(M, M 2,e )el }] ⎥ n ⎥. ⎢ =⎢ ⎥ 2,e )e }] −[n, rot rot{G(M, M ⎦ ⎣ l n 2,e −jkεe [n, rot{G(M, Mn )el }]
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Since ker Tˆ ∗ = {0}, it suffices to show that the relations [n, rot rotP {j0 (P, Mns,i )el }]es∗ (P ) dσP ∂Dis
[n, rotP {j0 (P, Mns,i )el }]hs∗ (P ) dσP = 0,
+ jkεis ∂Dis
(3s )
[n, rot rotP {G(P, Mns,e )el }]e1∗ (P ) dσP ∂Di1
s = 1, 2,
[n, rotP {G(P, Mns,e )el }]h1∗ (P ) dσP
+ jkεe ∂Di1
[n, rot rotP {G(P, Mns,e )el }]e2∗ (P ) dσP
+ ∂Di2
[n, rotP {G(P, Mns,e )el }]h2∗ (P ) dσP = 0,
+ jkεe
s = 1, 2,
(4s )
∂Di2
imply that {e1 , h1 , e2 , h2 } ∼ 0 in Hτ for arbitrary n and l. Let us prove this. Consider relation (31 ). Since {Mn1,i } = S1i , we have [n, rot rotP {j0 (P, M )el }]e1∗ (P ) dσP ∂Di1
[n, rotP {j0 (P, M )el }]h1∗ (P ) dσP = 0,
+ jkεi1
M ∈ S1i .
∂Di1
The function occurring on the left-hand side in the relation satisfies the Helmholtz equation Δu + ki21 u = 0 in Di1 and is zero on the normal surface S1i . This, together with the analyticity of the function, implies that [n, rot rotP {j0 (P, M )el }]e1∗ (P ) dσP ∂Di1
[n, rotP {j0 (P, M )el }]h1∗ (P ) dσP = 0,
+ jkεi1
M ∈ Di1 .
(5)
∂Di1
Since j0 (P, M )|M∈Di1 = j0 (ki1 RP M ) is an entire function, it follows that relation (5) holds in any finite domain R3 . In the domain De , we take a sphere Σ that is nonresonance for the given wave number ki1 and contains the entire Di1 inside itself. For j0 (ki1 RP M ), we use the representation [7, p. 157 of the Russian translation]
∞ l j −1 m N J (P )Jlm∗ (M ), j0 (P, M ) = 2π l =0 m=−l l m l
where Nl m =
1 l − m , 2l + 1 l + m
m imϕ Jlm . (P ) = jl (ki1 r)Pl (cos θ)e
We substitute the expansion of j0 (P, M ) into relation (5) on Σ and use the orthogonality of (M ) on Σ : Jlm∗ m 1∗ 1∗ [n, rot rotP {Jl (P )el }]e (P ) dσP + jkεi1 [n, rotP {Jlm (P ) dσP = 0. (P )el }]h ∂Di1
∂Di1
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−1 m By multiplying the resulting relation by Nlm Hl (Q), Q ∈ Σ, and by summing over l and m, we find that the function [n, rot rotP {ψi1 (P, M )el }]e1∗ (P ) dσP Ul (M ) = ∂Di1
[n, rotP {ψi1 (P, M )el }]h1∗ (P ) dσP ,
+ jkεi1
l = 1, 2, 3,
∂Di1
where ψi1 (P, M ) =
e−jki1 RP M , satisfies the boundary value problem RP M ΔUl + ki21 Ul = 0 outside Σ, Ul (Q) = 0, Q ∈ Σ, ∂Ul + jki1 Ur = o(r −1 ), r → ∞. ∂r
Hence we find that Ul (M ) ≡ 0 outside Σ, and Ul ≡ 0 in De by virtue of the analyticity of Ul . Therefore, [n, rot rotP {ψi1 (P, M )el }]e1∗ (P ) dσP ∂Di1
[n, rotP {ψi1 (P, M )el }]h1∗ (P ) dσP = 0,
+ jkεi1
M ∈ De .
(6)
M ∈ De ,
(7)
∂Di1
Likewise, for relation (32 ), we obtain the relation [n, rot rotP {ψi2 (P, M )el }]e2∗ (P ) dσP ∂Di2
[n, rotP {ψi2 (P, M )el }]h2∗ (P ) dσP = 0,
+ jkεi2 ∂Di2
where ψis (P, M ) =
e−jkis RP M , RP M
s = 1, 2.
By performing vector transformations, one can rewrite relations (6) and (7) in the form [es∗ (P ), n] rot rotP {ψis (P, M )el }dσP ∂Dis
[hs∗ (P ), n] rotP {ψis (P, M )el } dσP = 0,
+ jkεis
M ∈ De ,
s = 1, 2.
(8s )
∂Dis
We introduce the following notation: as (P ) ≡ [es∗ (P ), n],
bs (P ) ≡ [hs∗ (P ), n],
where a1 (P ), a2 (P ), b1 (P ), b2 (P ) ∈ Hτ . DIFFERENTIAL EQUATIONS
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s = 1, 2,
(9)
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BARYSHEV, EREMIN
Then relations (8s ) acquire the form rot rotP {ψis (P, M )el }as (P ) dσP ∂Dis
rotP {ψis (P, M )el }bs (P ) dσP = 0,
+ jkεis
M ∈ De ,
s = 1, 2.
(10)
∂Dis
By performing vector transformations by analogy with [4, p. 44], we can represent the integrals occurring in (10) in the form 1 rot rotP {ψi1 (P, M )el }a (P ) dσP = el · rot rotM ψi1 (P, M )a1 (P ) dσP , ∂Di1
∂Di1
rotP {ψi1 (P, M )el }b1 (P ) dσP = el · rotM ∂Di1
ψi1 (P, M )b1 (P ) dσP . ∂Di1
These relations represent the mutuality principle. As a result, relations (10) are equivalent to the relations ψis (P, M )as (P ) dσP rot rotM ∂Dis
ψis (P, M )bs (P ) dσP = 0,
+ jkεis rotM
M ∈ De ,
s = 1, 2.
(11)
∂Dis
Let us proceed to the analysis of relations (4). Consider the first of them. By virtue of the relation {Mn1,e } = S1e , we have [n, rot rotP {G(P, M )el }]e1∗ (P ) dσP + jkεe [n, rotP {G(P, M )el }]h1∗ (P ) dσP ∂Di1
∂Di1
[n, rot rotP {G(P, M )el }]e2∗ (P ) dσP
+ ∂Di2
[n, rotP {G(P, M )el }]h2∗ (P ) dσP = 0,
+ jkεe
M ∈ S1e .
(12)
∂Di2
The function occurring on the left-hand side in (12) satisfies the Helmholtz equation Δu + ki21 u = 0 in Di1 and vanishes on the normal surface Se1 . This, together with the analyticity of the function, implies that [n, rot rotP {G(P, M )el }]e1∗ (P ) dσP + jkεe [n, rotP {G(P, M )el }]h1∗ (P ) dσP ∂Di1
∂Di1
[n, rot rotP {G(P, M )el }]e2∗ (P ) dσP
+ ∂Di2
[n, rotP {G(P, M )el }]h2∗ (P ) dσP = 0,
+ jkεe
M ∈ Di1 .
(13)
∂Di2
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By making vector transformations by analogy with the case of relations (6) and (7) and by using notation (9), we rewrite the expression (13) in the form rot rotP {G(P, M )el }a1 (P ) dσP + jkεe rotP {G(P, M )el }b1 (P ) dσP ∂Di1
∂Di1
rot rotP {G(P, M )el }a2 (P ) dσP
+ ∂Di2
rotP {G(P, M )el }b2 (P ) dσP = 0,
+ jkεe
M ∈ Di1 .
(14)
∂Di2
By the mutuality principle, we obtain the relations rot rotP {G(P, M )el }a1,2 (P ) dσP ≡ el · rot rotM ∂Di1 ,i2
G(P, M )a1,2 (P ) dσP , ∂Di1 ,i2
rotP {G(P, M )el }b1,2 (P ) dσP ≡ el · rotM
∂Di1 ,i2
G(P, M )b1,2 (P ) dσP . ∂Di1 ,i2
Since the vector el is arbitrary, it follows that relation (14) acquires the form 1 G(P, M )a (P ) dσP + jkεe rotM G(P, M )b1 (P ) dσP rot rotM ∂Di1
∂Di1
2
+ rot rotM
G(P, M )b2 (P )dσP = 0,
G(P, M )a (P ) dσP + jkεe rotM ∂Di2
M ∈ Di1 .
(15)
M ∈ Di2 .
(16)
∂Di2
In a similar way, by transforming the second relation in (4), we obtain 1 G(P, M )a (P ) dσP + jkεe rotM G(P, M )b1 (P ) dσP rot rotM ∂Di1
∂Di1
G(P, M )a2 (P ) dσP + jkεe rotM
+ rot rotM ∂Di2
G(P, M )b2 (P ) dσP = 0, ∂Di2
We have thereby reduced the expressions (3s ) and (4) to the form ψis (P, M )as (P ) dσP rot rotM ∂Dis
ψis (P, M )bs (P ) dσP = 0,
+ jkεis rotM ∂Dis
∂Di1
s = 1, 2,
(17)
G(P, M )a1 (P ) dσP + jkεe rotM
rot rotM
M ∈ De ,
G(P, M )b1 (P ) dσP ∂Di1
G(P, M )a2 (P ) dσP
+ rot rotM ∂Di2
G(P, M )b2 (P ) dσP = 0,
+ jkεe rotM ∂Di2
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M ∈ Dis ,
s = 1, 2.
(18)
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BARYSHEV, EREMIN
By applying the rot operation to relations (17) and (18), we obtain ψis (P, M )bs (P ) dσP
rot rotM ∂Dis
− jkμis rotM ∂Dis
M ∈ De ,
ψis (P, M )as (P ) dσP = 0,
G(P, M )b (P ) dσP − jkμe rotM ∂Di1
(19)
1
rot rotM
s = 1, 2,
G(P, M )a1 (P ) dσP ∂Di1
G(P, M )b2 (P ) dσP
+ rot rotM ∂Di2
− jkμe rotM
M ∈ Dis ,
G(P, M )a2 (P ) dσP = 0,
s = 1, 2.
(20)
∂Di2
Note that the functions on the left-hand sides in (18) and in (20) are the same. We denote the functions occurring on the left-hand sides in relations (17) by E i1 and E i2 , respectively, and by E e we denote the function on the left-hand side in relations (18). Likewise, we introduce the notation E i1 ,i2 ,e for relations (19) and (20). From relations (17), (18) and (19), (20), we have
lim
Di1 M→Q∈∂Di1 De M→Q∈∂Di1
n,
E 1i − E e
H1i − He
= 0,
lim
Di2 M→Q∈∂Di2 De M→Q∈∂Di2
n,
E 2i − E e
H2i − He
= 0.
(21)
1,2 By taking into account the representations for E 1,2 i , E e , Hi , and He and the jumps of the vector potentials on ∂Di1 and ∂Di2 , for the first expression in (21), we obtain the relations
1 1 1 (μi + μf )a (Q) + nQ , rotQ {μf G(Q, P ) − μi1 ψi1 (Q, P )}a (P )dσP 2 1
1 nQ , rot rotQ − jk
∂Di1
{G(Q, P ) − ψi1 (Q, P )}b1 (P ) dσP
∂Di1
2
+ μf nQ , rotQ
G(Q, P )a (P ) dσP
∂Di2
1 (εi + εf )b1 (Q) + nQ , rotQ 2 1
1 nQ , rot rotQ + jk
+ εf nQ , rotQ ∂Di2
1 2 nQ , rot rotQ G(Q, P )b (P ) dσP = 0, − jk ∂Di2
(22)
{εf G(Q, P ) − εi1 ψi1 (Q, P )}b (P ) dσP 1
∂Di1
{G(Q, P ) − ψi1 (Q, P )}a1 (P ) dσP
∂Di1
2
G(Q, P )b (P ) dσP
1 2 nQ , rot rotQ G(Q, P )a (P ) dσP = 0. + jk ∂Di2
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For the second expression in (21), we have
1 2 2 (μi + μf )a (Q) + nQ , rotQ {μf G(Q, P ) − μi2 ψi2 (Q, P )}a (P ) dσP 2 2
1 nQ , rot rotQ − jk
∂Di2
{G(Q, P ) − ψi2 (Q, P )}b (P ) dσP
∂Di2
G(Q, P )a1 (P ) dσP
+ μf nQ , rotQ ∂Di1
1 (εi + εf )b2 (Q) + nQ , rotQ 2 2
1 nQ , rot rotQ + jk
2
+ εf nQ , rotQ
1 nQ , rot rotQ G(Q, P )b1 (P ) dσP = 0, − jk ∂Di1
{εf G(Q, P ) − εi2 ψi2 (Q, P )}b2 (P ) dσP
∂Di2
(23)
{G(Q, P ) − ψi2 (Q, P )}a (P ) dσP 2
∂Di2
G(Q, P )b1 (P ) dσP
1 nQ , rot rotQ G(Q, P )a1 (P ) dσP = 0 + jk
∂Di1
∂Di1
almost everywhere on ∂Di1 and ∂Di2 . The system of equations (22) and (23) is a Fredholm system of the second kind. Since there exists a solution {a1 , b1 , a2 , b2 } ∈ Hτ , we have {a1 , b1 , a2 , b2 } ∼ {A1 , B1 , A2 , B2 } ∈ Hτ1 ≡ Cτ1,α (∂Di1 ) × Cτ1,α (∂Di1 ) × Cτ1,α (∂Di2 ) × Cτ1,α (∂Di2 ). By virtue of the unique solvability of the original boundary value problem (1), system (22), (23) has only the trivial solution in Hτ1 , whence it follows that {a1 , b1 , a2 , b2 } ∼ 0. Then, by taking into account (9), we obtain {e1 , h1 , e2 , h2 } ∼ 0. The proof of the theorem is complete. For the diffraction problem (1), one can obtain the well-posedness relation 2 N N N 0 N N 0 Hiq − He − H Lτ2 (∂Diq ) + Eiq − Ee − E Lτ2 (∂Diq ) . Ee − Ee C(de,i ) = O q=1
By using this well-posedness relation and by taking into account the completeness and closedness (proved in Theorem 1) of the system of basis functions used in the representation of the approximate solution, we obtain the following main result. Theorem 2. Let all assumptions of Theorem 1 hold; then the approximate solution constructed on the basis of the discrete source method (2) converges to the exact solution of the boundary value diffraction problem (1), N lim {EN i1 ,i2 ,e (M ), Hi1 ,i2 ,e (M )} = {Ei1 ,i2 ,e (M ), Hi1 ,i2 ,e (M )},
N →∞
M ∈ de,i ,
where de,i are closed bounded domains in De,i .
ACKNOWLEDGMENTS The research was supported by the Russian Foundation for Basic Research (project no. 09-0100318). DIFFERENTIAL EQUATIONS
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DIFFERENTIAL EQUATIONS
Vol. 47
No. 9
2011
