ISSN 10645624, Doklady Mathematics, 2011, Vol. 84, No. 2, pp. 613–616. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V.G. Romanov, 2011, published in Doklady Akademii Nauk, 2011, Vol. 440, No. 1, pp. 21–24.

MATHEMATICS

The Problem of Determining the Kernel of Electrodynamics Equations for Dispersion Media Corresponding Member of the Russian Academy of Sciences V. G. Romanov Received April 28, 2011

DOI: 10.1134/S1064562411060068

The propagation of electromagnetic waves in dis persion media is described by the equations ∂ ⎛ ε  ⎜ 0 ( x )E ( x, t ) + ∂t ⎝

t

⎞

–∞

(1)

– curlH ( x, t ) + j ( x, t ) = 0, 4

( x, t ) ∈ ⺢ .

In these equations, ε0(x) is the dielectric permeability of the medium and the coefficient ε(x, t) characterizes the medium dispersion. The convolution k * E corre sponds to a certain “memory” of the medium. In what follows, we consider the system of equations (1) under the zero initial conditions ( E, H ) t < 0 = 0

∑

(2)

with foreign current having the form of a moment dipole concentrated at a point y ∈ ⺢3 and having direction j 0; i.e., j = j 0δ(x – y, t). It is natural to assume that j 0 ≠ 0. We also assume that the function ε(x, t) can be represented in the form ε(x, t) = k(t)ε0(x)p(x), where k(0) = 1 and k(t) is a known function. Suppose that the supports of the functions ε0(x) – 1 and p(x) are con tained in a compact open domain Ω ⊂ ⺢3 with smooth boundary ∂Ω . We are interested in determining a pair of functions ε0(x) and p(x) from a certain information about the solution of problem (1), (2). We give an exact setting of this inverse problem a little latter. 2

∂ ln ε 0 ( x ) ν i ν j ≥ 0, ∀: ν = ( ν 1, ν 2, ν 3 ) ≠ 0, ∂x i ∂x j

x ∈ Ω. Suppose that, in addition, the domain Ω is convex with respect to geodesics. Under these conditions, the metric is simple in the closed domain Ω ; i.e., any two points x and y in Ω can be joined by a unique geodesic. We denote the geodesic line joining points x and y by Γ(x, y) and its Riemannian length by τ(x, y). Let η > 0 be any number. Consider the following inverse problem. Suppose that the function H(x, t, y) is given for all (x, y) ∈ (∂Ω × ∂Ω) and all t ≤ τ(x, y) + η, i.e., H ( x, t, y ) = f ( x, t, y ), ( x, y ) ∈ ( ∂Ω × ∂Ω ), (3) t ≤ τ ( x, y ) + η. It is required to determine ε0(x) and p(x) on Ω from the function f(x, t, y). Below, we give some arguments which reduce this problem to simpler problems to be solved successively. In the case where ε0(x) ≡ 1 and ε(x, t) ≡ 0, the solu tion of problem (1), (2) has the form

Suppose that the Riemannian metric dτ = 2

2

i, j = 1

∫ ε ( x, t – s )E ( x, s ) ds⎟⎠

∂  H ( x, t ) + curlE ( x, t ) = 0; ∂t

3

2

ε 0 ( x ) ( dx 1 + dx 2 + dx 3 ) has nonpositive curvature in Ω . A sufficient condition for this is the inequality

0

j H ( x, t ) = curl   δ(t – x – y ) 4π x – y 0

j ×ν 1 =  δ' ( t – x – y ) + δ ( t – x – y ) , 4π x – y x–y 0

∂ j E ( x, t ) = –   δ ( t – x – y ) ∂t 4π x – y 0

j + ∇div   θ0 ( t – x – y ) 4π x – y Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090 Russia email: [email protected]

1 ⎧ ν(ν ⋅ j ) – j =  ⎨ δ' ( t – x – y ) 4π ⎩ x–y 0

613

0

(4)

614

ROMANOV 0

0

( ν ⋅ j ) – jδ ( t – x – y ) + 3ν  2 x–y 0 0 ⎫ 3ν ( ν ⋅ j ) – j +  θ 0 ( t – x – y ) ⎬, 3 x–y ⎭

x – y and θ (t) is the Heaviside function. where ν =  0 x–y In what follows, we assume that the point y ∈ ∂Ω is a variable parameter of the problem, and j0 = j0(y) is a nonzero vector in the tangent plane to ∂Ω at the point y. Lemma 1. If the functions ε0(x) and ε(x, t) satisfy the assumptions made above and are sufficiently smooth (say, of class C∞(Ω × ⺢)), then, for small η > 0, the solu tion of problem (1), (2) can be represented is a form sim ilar to (3); namely, H ( x, t, y ) = α H ( x, y )δ ' ( t – τ ( x, y ) ) ˆ ( x, t, y ), + β ( x, y )δ ( t – τ ( x, y ) ) + H H

E ( x, t, y ) = α E ( x, y )δ' ( t – τ ( x, y ) ) ˆ ( x, t, y ), + β ( x, y )δ ( t – τ ( x, y ) ) + E

(5)

E

t ≤ τ ( x, y ) + η. Here, αH(x, y), βH(x, y), αE(x, y), and βE(x, y) are solu tions of the equations ( 2∇τ ( x, y ) ⋅ ∇ + Δτ ( x, y ) + ε 0 ( x )p ( x ) )α H ( x, y ) – ∇ε 0 ( x ) × α E ( x, y ) = 0, ( 2∇τ ( x, y ) ⋅ ∇ + Δτ ( x, y ) + ε 0 ( x )p ( x ) )β H ( x, y )

(6)

– Δα H ( x, y ) + k' ( 0 )ε 0 ( x )p ( x )α H ( x, y )

(9) H ˆ Here, f (x, t, y) is the regular part of the function f(x, t, y). Note that αH(x, y) ≠ 0 for all (x, y) ∈ (∂Ω × ∂Ω) and x ≠ y (see the corollary of Lemma 2 stated below). Let us fix points x ∈ ∂Ω and y ∈ ∂Ω . Consider the function f (x, t, y) as a function of the variable t. This function identically vanishes at t < τ(x, y) and has nonzero sin gular part at t = τ(x, y), because αH(x, y) ≠ 0. Therefore, τ(x, y) = sup{τ}, {τ} = {τ ∈ ⺢ | u(x, t, y) = 0 if t < τ}. Note that α H ( x, y ) =

ε 0 ( x )α E ( x, y ) + ∇τ ( x, y ) × α H ( x, y ) = 0, (7)

× β H ( x, y ) – rotα H ( x, y ) = 0, and satisfy the limit conditions 0

j (y) × ν(y) lim [ α H ( x, y )τ ( x, y ) ] =  , x→y 4π

f ( x, t, y ) = α H ( x, y )δ' ( t – τ ( x, y ) ) + β ( x, y )δ ( t – τ ( x, y ) ) + ˆf ( x, t, y ).

t

– ∇p ( x ) × α E ( x, y ) + ∇ε 0 ( x ) × β E ( x, y ) = 0, ε 0 ( x )β E ( x, y ) + ε 0 ( x )p ( x )α E ( x, y ) + ∇τ ( x, y )

the function H(x, t, y) and consider the system of this secondorder equation for H(x, t, y) and a firstorder equation for E(x, t, y). Substituting representation (5) into this system, we find relations (6) and (7). By virtue of the assumptions made above, in some neighbor hood of y ∈ ∂Ω, we have ε0(x) = 1 and p(x) = 0. There fore, for each fixed point y ∈ ∂Ω, formula (4) uniquely determines the values of the functions αH(x, y), βH(x, y), αE(x, y), and βE(x, y) in some neighborhood of this point. This implies the limit relations (8). As a result, the values of the functions αH(x, y) and βH(x, y) inside Ω are constructed, after eliminating αE(x, y) and βE(x, y) from Eqs. (6) by using algebraic relations (7), along geodesics as solutions of ordinary differential equa tions, after which the functions αE(x, y) and βE(x, y) are found in explicit form. The system of equations for ˆ (x, t, y), E ˆ (x, t, y), which results from substituting H representation (5), makes it possible to prove the reg ularity of these functions. It follows from (5) that

(8) 0 j ( y ) × ν ( y ) lim [ β H ( x, y )τ ( x, y ) ] = , x→y 4π as x tends to y along the geodesic Γ(x, y); here, ν(y) is the unit tangent vector to Γ(x, y) at the point y. The functions ˆ (x, t, y) and E ˆ (x, t, y) are certain functions of the vari H ables (x, t) which are regular at x ≠ y and, moreover, sat ˆ (x, t, y) = 0 and E ˆ (x, t, y) = 0 for all isfy the conditions H t < τ(x, y). Below we briefly outline the proof of this lemma. It is convenient to construct a secondorder equation for

lim

t → τ ( x, y ) + 0

∫ ( t – s )f ( x, s, y ) ds,

–∞

(10)

( x, y ) ∈ ( ∂Ω × ∂Ω ). Thus, the inverse problem stated above can be refor mulated as follows: determine ε0(x) and p(x) inside Ω from functions τ(x, y) and αH(x, y) given for all (x, y) ∈ (∂Ω × ∂Ω). This problem, in turn, reduces to the fol lowing two problems to be solved successively. Problem 1. Determine ε0(x) inside Ω from a func tion τ(x, y) given for any pair of points x, y belonging to ∂Ω . Problem 2. For a known function ε0(x), determine p(x) inside Ω from a function αH(x, y) given at (x, y) ∈ (∂Ω × ∂Ω). Problem 1 is called the inverse kinematic problem and has been well studied (see, e.g., books [1, 2]). The stability of its solution was estimated in the case of a twodimensional space by Mukhometov in [3] and in the case of a higherdimensional space, in [4–7]. Here, we consider Problem 2. DOKLADY MATHEMATICS

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THE PROBLEM OF DETERMINING THE KERNEL

Lemma 2. The function |αH(x, y)| has the expression α H ( x, y ) j ( y ) × ν ( y ) det ⎛ ∂ζ ⎞ ⎝ ∂x⎠ ⎛ 1 =  exp ⎜ –  4πτ ( x, y ) ⎝ 2 0

∫

Γ ( x, y )

⎞ p ( ξ ) dτ⎟ , (11) ⎠

in which ζ = (ζ1, ζ2, ζ3) = ν(y)τ(x, y) is the vector of Riemann coordinates of the point x and det ⎛ ∂ζ ⎞ is the ⎝ ∂x⎠ Jacobian of the transition from the Riemann coordinates to the Cartesian ones. The proof of this lemma is fairly simple; we give it here, because formula (11) plays the fundamental role in the problem under consideration. Equalities (6) and (7) imply a differential equation for the function αH. It has the form ( 2∇τ ( x, y ) ⋅ ∇ + Δτ ( x, y ) + ε 0 ( x )p ( x ) )α H ( x, y ) + ∇ ln ε 0 ( x ) × ( ∇τ ( x, y ) × α H ( x, y ) ) = 0.

(12)

This equation readily implies αH(x, y) · ∇τ(x, y) = 0. As is well known, the principal singularities of the vectors H and E are oriented in tangent directions to the front of the electromagnetic wave propagation. Using this fact and taking the inner products of both sides equality (12) and αH(x, y), we obtain an equation for |αH(x, y)|2 of the form 2

∇τ ( x, y ) ⋅ ∇ α H ( x, y ) + ( Δτ ( x, y ) + ε 0 ( x )p ( x ) – ∇τ ( x, y ) ⋅ ∇ ln ε 0 ( x ) ) α H ( x, y )

2

= 0.

4πτ ( x, y ) α H ( x, y ) p ( ξ ) dτ = – 2 ln   ≡ g ( x, y ), 0 ∂ζ (15) Γ ( x, y ) j ( y ) × ν ( y ) det ⎛ ⎞ ⎝ ∂x⎠

∫

( x, y ) ∈ ( ∂Ω × ∂Ω ), which reduces Problem 2 under consideration to an integral geometry problem studied in [1–8]. Below, we give a result related to the stability of its solution, fol lowing [2]. Suppose that a function x = χ(ξ) implements a onetoone mapping of class C2 from the unit sphere S2 centered at the origin to ∂Ω so that the positive ori entation of ∂Ω corresponds to the positive orientation of S2. Suppose also that θ and ϕ are the spherical coor dinates of the point ξ and ϒ = [0, π] × [0, 2π] is the range of variation of the variables θ and ϕ. Finally, sup pose that points x ∈ ∂Ω and y ∈ ∂Ω are mapped to points ξ(θ1, ϕ1) and ξ(θ2, ϕ2), respectively. Then x = χ(ξ(θ1, ϕ1)) ≡ x(θ1, ϕ1) and y = χ(ξ(θ2, ϕ2)) ≡ y(θ2, ϕ2). We set g(x(θ1, ϕ1), y(θ2, ϕ2)) ≡ g (θ1, ϕ1, θ2, ϕ2). Then relation (15) can be written in the form

∫

(13)

in which d denotes differentiation along the geodesic dτ d Γ(x, y). Observing that ∇τ(x, y) · ∇ = ε0(x)  , we find dτ

( ( θ 1, ϕ 1 ), ( θ 2, ϕ 2 ) ) ∈ ϒ × ϒ.

by I( g , τ ) we denote the determinant

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⎛ ⎜ 0 g θ1 g ϕ1 I ( g, τ ) = det ⎜ g θ τ θ θ τ ϕ θ ⎜ 2 1 2 1 2 ⎜ ⎝ g ϕ2 τ θ1 ϕ2 τ ϕ1 ϕ2

⎞ ⎟ ⎟. ⎟ ⎟ ⎠

Theorem. If ∂Ω ∈ C2, ε0(x) ∈ C2(Ω ∪ ∂Ω), p(x) ∈ C1(Ω ∪ ∂Ω), and the family of geodesics is regular in Ω , then the stability of the solution is estimated as

∫ p ( x )ε 2

Ω

3/2 0 ( x ) dx

1 ≤ –  8π

∫

I ( g, τ ) dθ 1 dϕ 1 dθ 2 dϕ 2 . (17)

ϒ×ϒ

(14) ACKNOWLEDGMENTS

Using the limit equality (8) for the function αH(x, y), we obtain formula (11). Corollary. For all (x, y) ∈ ∂Ω × ∂Ω such that x ≠ y, αH(x, y) ≠ 0. Formula (11) implies the relation DOKLADY MATHEMATICS

(16)

We also set τ(x(θ1, ϕ1), y(θ2, ϕ2)) ≡ τ (θ1, ϕ1, θ2, ϕ2);

2  –  d ln det ⎛ ∂ζ = ε 0 ( x )  ⎞ , ⎝ ∂x⎠ τ ( x, y ) dτ

∫

p ( ξ ) dτ = g ( θ 1, ϕ 1, θ 2, ϕ 2 ),

Γ ( x ( θ 1, ϕ 1 ), y ( θ 2, ϕ 2 ) )

1 Using formula (2.2.35) in [2] with aij =  , we ε0 ( x ) obtain Δτ ( x, y ) – ∇τ ( x, y ) ⋅ ∇ ln ε 0 ( x )

⎛ ⎞ ⎜ α H ( x, y ) 2 τ 2 ( x, y ) ⎟ d  ⎜  exp p ( ξ ) dτ⎟ = 0. dτ ⎜ ∂ζ ⎟ det ⎛ ⎞ Γ ( x, y ) ⎝ ⎠ ⎝ ∂x⎠

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2011

This work was supported by the Russian Founda tion for Basic Research (project no. 110100105a), the Ministry of Education and Science of Russian Federation (State Contract no. 14.740.11.0350), and Siberian Division, Russian Academy of Sciences (ExternalAid2009 Project no. 93).

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REFERENCES 1. V. G. Romanov, Inverse Problems of Mathematical Physics (Nauka, Moscow, 1984; VNU Science Press, Utrecht, 1987). 2. V. G. Romanov, Stability in Inverse Problems (Nauchnyi Mir, Moscow, 2005) [in Russian]. 3. R. G. Mukhometov, Dokl. Akad. Nauk SSSR 232 (1), 32–35 (1977). 4. R. G. Mukhometov and V. G. Romanov, Dokl. Akad. Nauk SSSR 243 (1), 41–44 (1978).

5. I. N. Bernshtein and M. L. Gerver, Dokl. Akad. Nauk SSSR 243 (2), 302–305 (1978). 6. I. N. Bernshtein and M. L. Gerver, in Methods and Algorithms for Interpretation of Seismic Data (Nauka, Moscow, 1980) [in Russian]. 7. G. Ya. Beil’kin, in BoundaryValue Problems of Mathe matical Physics and Related Questions of Function The ory (Nauka, Leningrad, 1979), pp. 3–6 [in Russian]. 8. V. G. Romanov, Dokl. Akad. Nauk SSSR 241 (2), 290– 293 (1978).

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2011