Journal of Mathematical Sciences, Vol. 193, No. 2, August, 2013

BOUNDARY VALUE PROBLEM FOR THE RADIATION TRANSFER EQUATION WITH DIFFUSE REFLECTION AND REFRACTION CONDITIONS A. A. Amosov National Research University “Moscow Power Engineering Institute” 14, Krasnokazarmennaya St., Moscow 111250, Russia [email protected]

UDC 517.95

We study the boundary value problem for the radiation transfer equation with diffuse reflection and refraction conditions. We establish the existence and uniqueness of a solution. We also obtain a priori estimates for the solution. Bibliography: 12 titles.

Introduction This paper continues the study started by the author in [1], where boundary value problems for the radiation transfer equation with reflection and refraction conditions in accordance with the Fresnel laws were considered. Unlike [1], the case of diffuse reflection and refraction conditions is studied in this paper. We consider the problem of monochormatic radiation transfer in the system G = ∪m j=1 Gj of semitransparent bodies Gj separated by the vacuum. Each body Gj is a bounded domain in R3 with smooth boundary ∂Gj of class C 1 . We assume that the domains Gi and Gj are pairwise disjoint, but for some i = j the boundaries of Gi and Gj can intersect. Let Ω = {ω ∈ R3 | |ω| = 1} be the unit sphere in R3 (the sphere of directions). The sought function I(ω, x) is defined on the set D = Ω × G and is interpreted as the radiation intensity at a point x ∈ G when the radiation propagates along a direction ω ∈ Ω. We assume that each body Gj is occupied by a semitransparent medium with constant absorption κj > 0, scattering coefficient sj  0, and refraction exponent kj > 1. We set κ(x) = κj , s(x) = sj , and k(x) = kj for x ∈ Gj , 1  j  m. To describe the radiation propagation in G, we use the equation of radiation transfer ω · ∇I + βI = sS (I) + κk 2 F, where ω · ∇I =

3 

ωi

i=1

∂ I is the derivative of a function I in a direction ω and S denotes the ∂xi

scattering operator S (I)(ω, x) =

(ω, x) ∈ D,

1 4π



θj (ω  · ω)I(ω  , x) dω  ,

(ω, x) ∈ Dj = Ω × Gj ,

1  j  m,

Ω

Translated from Problemy Matematicheskogo Analiza 71, June 2013, pp. 3–26. c 2013 Springer Science+Business Media New York 1072-3374/13/1932-0151 

151

with the scattering indicatrix possessing the following properties: 1

θj ∈ L (−1, 1),

θj  0,

1 2

1 θj (μ) dμ = 1,

1  j  m.

−1

Furthermore, β(x) = κ(x) + s(x) is the extinction coefficient and F (ω, x) characterizes the density of radiation of volume sources. The most studied problems in this relation are boundary value problems for the equation of radiation transfer with the continuity condition for the radiation intensity imposed on the interface between media with different optic properties, i.e., the radiation passes through the interface without changing direction and intensity. In the case of nonconvex domains, this condition leads to the “shooting condition” [2]. However, in many applications (for example, optics, tomography, thermal physics), the reflection and refraction of radiation at the interface between media should be taken into account. Some problems with conditions taking into account reflection on the interface between media were treated in [3, 4]. From the mathematical point of view, problems for the equation of radiation transfer with reflection and refraction conditions in accordance with the Fresnel laws were first studied in the papers [5, 6, 1]. In this paper, we study the boundary value problem for the equation of radiation transfer in a system of semitransparent bodies separated by the vacuum with the conditions of diffuse reflection and diffuse refraction at the interface between media. We essentially use the results obtained in the previous paper by the author [1] and the methods developed there. The paper is organized as follows. In Section 1, we prove a number of auxiliary assertions. In Section 2, we introduce necessary function spaces and discuss their properties. In Section 3, we recall some laws of diffuse reflection and diffuse refraction. We also give a preliminary statement of the conditions of diffuse reflection and refraction. Section 4 is devoted to the study of properties of the boundary operators used in this paper. Finally, in Section 5, we formulate the boundary value problem, derive a priori estimates for solutions and prove the existence and uniqueness results.

1

Auxiliaries

We regard R3 as an Euclidean space with elements x = (x1 , x2 , x3 ) equipped with the inner 3  xi yi . Let ω0 ∈ Ω. We denote by πω0 the plane πω0 = {y ∈ R3 | ω0 · y = 0} product x · y = i=1

with normal vector ω0 passing through the origin and the orthonormal basis e1 (ω0 ), e2 (ω0 ). We denote by Pω0 the operator of orthogonal projection onto the plane πω0 . For ω0 ∈ Ω m  ∂Gj we consider the rays − (ω0 , x0 ) = {x = x0 − tω0 | t > 0} and and x0 ∈ ∂G = j=1

+ (ω0 , x0 ) = {x = x0 + tω0 | t > 0} emitted from the point x0 in the directions −ω0 and ω0 respectively. Open and closed disks in R2 and πω0 with center at the origin and radius r > 0 are denoted by Vr = {y  ∈ R2 | |y  | < r}, Vr,ω0 = {y ∈ πω0 | |y| < r},

V r = {y  ∈ R2 | |y  |  r}, V r,ω0 = {y ∈ πω0 | |y|  r}.

Hereinafter, y  = (y1 , y2 ) and y = y1 e1 (ω0 ) + y2 e2 (ω0 ). 152

Let a function γ be defined on V r,ω0 , and let γ (y  ) = γ(y) = γ(y1 e1 (ω0 ) + y2 e2 (ω0 )). We  ∈ C 1 (V r ). This property is invariant under the choice of an orthonormal write γ ∈ C 1 (V r,ω0 ) if γ bases e1 (ω0 ), e2 (ω0 ) in the plane πω0 . For α, β, γ ∈ C 1 (V r,ω0 ) and α < β we consider the surface Πr,γ (ω0 , x0 ) = {x = x0 + y + γ(y)ω0 | y ∈ Vr,ω0 } and curvilinear cylinder Cr,α,β (ω0 , x0 ) = {x = x0 + y + t ω0 | y ∈ Vr,ω0 , α(y) < t < β(y)} with the axis (ω0 , x0 ) = {x = x0 + tω0 | t ∈ R} and bounding surfaces Πr,α (ω0 , x0 ) and Πr,β (ω0 , x0 ). Throughout the paper, we assume that ∂Gj ∈ C 1 , 1  j  m. The assumption ∂Gj ∈ C 1 means that for any x0 ∈ ∂Gj and ω0 = nj (x0 ) (where nj is the outward normal to ∂Gj ) there exist numbers r0 > 0, h0 > 0 and a function γ ∈ C 1 (V r0 ,ω0 ), −h0 < γ < h0 , such that Gj ∩ Cr0 ,−h0 ,h0 (ω0 , x0 ) = Cr0 ,−h0 ,γ (ω0 , x0 ), (R3 \ Gj ) ∩ Cr0 ,−h0 ,h0 (ω0 , x0 ) = Cr0 ,γ,h0 (ω0 , x0 ), ∂Gj ∩ Cr0 ,−h0 ,h0 (ω0 , x0 ) = Πr0 ,γ (ω0 , x0 ); (0)| = 0, where moreover, γ (0) = 0, |∇ γ 





(y ) = ∇γ

 ∂ γ  ∂ γ  (y ), (y ) . ∂y1 ∂y2

Some results will be obtained under the additional assumption that ∂Gj ∈ C 1+λ , where 0 < λ < 1. This assumption means that ∂Gj ∈ C 1 ; moreover, the constants r0 > 0 and h0 > 0 are independent of x0 and there exists a constant Aλ > 0 such that for each point x0 ∈ ∂Gj the corresponding function γ satisfies the H¨ older condition |γ(y1 ) − γ(y2 )|  Aλ |y1 − y2 |λ

∀ y1 , y2 ∈ V r0 ,ω0 .

Γj = Ω × ∂Gj ,

m

(1.1)

We set

Γ− j

Γ = Ω × ∂G = ∪ Γj , j=1

= {(ω, x) ∈ Γj | ω · nj (x) < 0},

Γ+ j = {(ω, x) ∈ Γj | ω · nj (x) > 0}, Γ0j = {(ω, x) ∈ Γj | ω · nj (x) = 0},

m

Γ− = ∪ Γ− j ,

j=1 m Γ+ = ∪ Γ+ , j=1 j m Γ0 = ∪ Γ0j . j=1

We recall properties of special subsets of Γ+ and Γ− and refer to [1] for a detailed exposition. ± 0 0 We note that Γ± j and Γ are open sets, whereas Γj and Γ are closed sets in the topology of the + set Γ. It is easy to see that (ω, x) ∈ Γ− j if and only if (−ω, x) ∈ Γj . Denote by dω and dσ(x) the measures induced by the Lebesgue measure in R3 on Ω and ∂G respectively. Assume that the measure dΓ(ω, x) = dω dσ(x) is defined on Γ and introduce on Γ− and Γ+ the measures − (ω, x) = |ω · nj (x)| dωdσ(x), dΓ + (ω, x) = ω · nj (x) dωdσ(x), dΓ

(ω, x) ∈ Γ− j , (ω, x) ∈ Γ+ j ,

1  j  m, 1  j  m.

For (ω, x) ∈ Dj ∪ Γ− j we set τ + (ω, x) = sup{t > 0 | x + sω ∈ Gj ∀s ∈ (0, t)}, + (ω, x) = x + τ + (ω, x)ω X

(1.2)

153

+ (ω, x) ∈ ∂Gj and (ω, X + (ω, x)) ∈ Γ+ ∪ Γ0 . and note that X j j For (ω, x) ∈ Dj ∪ Γ+ j we set

τ − (ω, x) = sup{t > 0 | x − sω ∈ Gj ∀s ∈ (0, t)}, − (ω, x) = x − τ − (ω, x)ω X

(1.3)

− (ω, x)) ∈ Γ− ∪ Γ0 . − (ω, x) ∈ ∂Gj and (ω, X and note that X j j We recall that ∂Gi and ∂Gj can intersect for some i = j and introduce the sets m

Σ=

Σj ,

Σj =

j=1

m

(∂Gj ∩ ∂Gi ),

1  j  m.

i=1 i=j

On ∂G \ Σ, we define the outward normal by setting n(x) = nj (x) for x ∈ ∂Gj \ Σj , 1  j  m. We also introduce the sets Sj− = {(ω, x) ∈ Γ− j | x ∈ ∂Gj \ Σj }, ∗

Sj+ = {(ω, x) ∈ Γ+ j | x ∈ ∂Gj \ Σj }, ∗

− − S− j = {(ω, x) ∈ Sj , |  (ω, x) ∩ G = ∅},

S− =

m

Sj− ,

S+ =

j=1

m

Sj+ ,

j=1

∗

S− =

m

+ + S+ j = {(ω, x) ∈ Sj |  (ω, x) ∩ G = ∅}, ∗

S− j ,

∗

S+ =

j=1

m

∗

S+ j .

j=1

∗

The set S − consists of (ω, x) ∈ S − such that the ray − (ω, x) does not intersect G. Similarly, ∗

the set S + consists of (ω, x) ∈ S + such that the ray + (ω, x) does not intersect G. ∗

Let (ω, x) ∈ S + \ S + . Then the ray + (ω, x) intersects G. We set τ + (ω, x) = inf {t > 0 | x + t ω ∈ G}, X + (ω, x) = x + τ + (ω, x)ω. It is clear that τ + (ω, x) > 0, X + (ω, x) ∈ ∂G, and (ω, X + (ω, x)) ∈ Γ− ∪ Γ0 . We set ∗

S+ = {(ω, x) ∈ S + \ S + | (ω, X + (ω, x)) ∈ Γ− }, ◦

∗

S + = {(ω, x) ∈ S + \ S + | (ω, X + (ω, x)) ∈ Γ0 }. ∗

Similarly, let (ω, x) ∈ S − \ S − . Then the ray − (ω, x) intersects G. We set τ − (ω, x) = inf {t > 0 | x − t ω ∈ G}, X − (ω, x) = x − τ − (ω, x)ω. It is clear that τ − (ω, x) > 0, X − (ω, x) ∈ ∂G, and (ω, X − (ω, x)) ∈ Γ+ ∪ Γ0 . We set ∗

S− = {(ω, x) ∈ S − \ S − | (ω, X − (ω, x)) ∈ Γ+ }, ◦

∗

S − = {(ω, x) ∈ S − \ S − | (ω, X − (ω, x)) ∈ Γ0 }. Lemma 1.1 (cf. [1]). The sets S+ and S− are open (in the topology of the set Γ), and the mapping (ω, x) → (ω, X + (ω, x)) is a homeomorphism from S+ onto S− with the inverse (ω, x) → (ω, X − (ω, x)). ◦

Lemma 1.2 (cf. [1]). meas (S ± ; dΓ) = 0. 154

2

Function Spaces

2.1. Lp (D) and W p (D). We recall that D = Ω × G and Dj = Ω × Gj , 1  j  m. It is m  Dj . Let 1  p  ∞. We denote by Lp (Dj ) the Banach space of functions f clear that D = j=1

defined on Dj and measurable with respect to the measure dω dx with finite norm ⎧  1 p ⎪ p ⎪ ⎪ |f (ω, x)| dωdx , 1  p < ∞, ⎪ ⎨ Dj

f Lp (Dj ) = ⎪ ⎪ ⎪ ess sup |f (ω, x)|, ⎪ p = ∞. ⎩ (ω,x)∈Dj

By the weak directional derivative of a function f ∈ L1 (Dj ) in a direction ω we mean a function w ∈ L1 (Dj ), denoted by w = ω · ∇f , satisfying the integral identity    f (ω, x) ω · ∇ϕ(ω, x) + w(ω, x)ϕ(ω, x) dω dx = 0 Dj

for all ϕ ∈ C(Dj ) such that ϕ(ω, ·) ∈ C0∞ (Gj ) for almost all ω ∈ Ω. We denote by W p (Dj ) the Banach space of functions f ∈ Lp (Dj ) possessing the weak derivatives ω · ∇f ∈ Lp (Dj ), equipped with the norm ⎧ 1 p ⎪ p ⎨ f p p +

ω · ∇f

, 1  p < ∞, L (Dj ) Lp (Dj )

f W p (Dj ) = ⎪ ⎩ max{ f L∞ (Dj ) , ω · ∇f L∞ (Dj ) }, p = ∞. For functions defined in D we introduce similar spaces Lp (D) and W p (D). It is clear that f ∈ Lp (D) if and only if f ∈ Lp (Gj ) for all 1  j  m and f ∈ W p (D) if and only if f ∈ W p (Dj ) for all 1  j  m. We also use the weighted spaces Lpρ (D), 1  p < ∞, with positive weight ρ ∈ L∞ (G) and the norm  1 p p |f (ω, x)| ρ(x) dωdx .

f Lpρ (D) = D

Let C (0,1) (Dj ) be the set of continuous functions ϕ defined in Dj and possessing continuous ∂ϕ , 1  i  3, in Dj . It is known (cf., for example, [11, 10]) that for partial derivatives ∂xi 1  p < ∞ the set C (0,1) (D j ) is dense in W p (Dj ). 2.2. Spaces of functions defined in Γ+ and Γ− . Let E ± be a subset of Γ± , measurable with respect to the measure dΓ. We denote by M(E ± ) the set of functions defined in E ± and p (E ± ) the Banach measurable with respect to the measure dΓ. We denote by Lp (E ± ) and L spaces of functions g ∈ M(E ± ) with finite norms ⎧  1 ⎪ p ⎪ p ⎪ ⎪ |g(ω, x)| dΓ(ω, x) , 1  p < ∞, ⎨

g Lp (E ± ) = E± ⎪ ⎪ ⎪ ess sup |g(ω, x)|, p = ∞, ⎪ ⎩ ± (ω,x)∈E

155

and

g Lp (E ± ) =

⎧  1 ⎪ p ⎪ p ± ⎪ ⎪ |g(ω, x)| dΓ (ω, x) , ⎨

1  p < ∞,

E±

⎪ ⎪ ⎪ ⎪ ⎩ ess sup± |g(ω, x)|,

p = ∞,

(ω,x)∈E

∞ (E ± ) = L∞ (E ± ). respectively. It is clear that L We introduce the space Lploc (Γ± ) as the set of all functions g ∈ M(Γ± ) such that g ∈ Lp (K) p (Γ± ) ⊂ Lp (Γ± ) for all 1  p < ∞. for any compact subset K ⊂ Γ± . It is clear that Lp (Γ± ) ⊂ L loc 1,p (E ± ) of functions g ∈ M(E ± ) that, after extending them We also use anisotropic spaces L by zero to Γ± \ E ± , possess the finite norms ⎧ m  1 p    p ⎪ ⎪ ⎪ |g(ω, x)||ω · nj (x)| dω dσ(x) , ⎪ ⎪ ⎪ ⎨ j=1 ∂Gj Ω± (x) j 

g L1,p (E ± ) = ⎪ ⎪ ⎪ |g(ω, x)||ω · nj (x)| dω, max ess sup ⎪ ⎪ ⎪ ⎩1jm x∈∂Gj ±

1  p < ∞, p = ∞.

Ωj (x)

− Hereinafter, Ω+ j (x) = {ω ∈ Ω | ω · nj (x) > 0} and Ωj (x) = {ω ∈ Ω | ω · nj (x) < 0}. Suppose that E ± ⊂ S ± and ν ∈ L∞ (∂G \ Σ), ν  0. We also need the weighted spaces 1 p ± ± ν (E ± ) and L 1,p p L ν (E ) of functions g ∈ M(E ) with finite seminorms g L  pν (E ± ) = ν g L  p (E ± ) 1

p and g L1,p ± = ν g L  1,p (E ± ) . ν (E )

 2.3. Traces of functions in W p (D). We recall how to define the traces f Γ± of a function m    on Γ± for f ∈ W p (D), 1  p  ∞, on Γ± . Since Γ± = Γ± j , it suffices to define the traces f Γ± j j j=1  every j, 1  j  m. For f ∈ C (0,1) (D j ) the traces f Γ± are naturally defined as the restrictions j

± ± of a function f to Γ± j . Let 1  p < ∞, and let K be an arbitrary compact subset of Γj . Then the following estimate holds (cf. [11]):  (2.1)

f Γ± Lp (K ± )  cK ± f W p (Dj ) ∀f ∈ C (0,1) (D j ), j

where the constant cK ± depends only on K ± , Gj , and p. Since the set C (0,1) (D  j ) is dense in W p (Dj ), the estimate (2.1) allows us to extend the linear operators f → f Γ± to linear continuous operators acting from W p (Dj ) to Lp (K ± ) and satisfying the estimate 

f Γ+ Lp (K ± )  cK ± f W p (Dj ) ∀f ∈ W p (Dj ).

j

j

Since Γ± be represented as the countable unions of sequences of expanding compact sets, j can  p  the traces f Γ+ ∈ Lploc (Γ± j ) are defined for f ∈ W (Dj ), 1  p < ∞. As a consequence, for any j  function f ∈ W p (D), 1  p < ∞, the traces f Γ± ∈ Lploc (Γ± ) are defined.  Remark 2.1. It is easy to see that for 1 < p < ∞ the linear trace operators f → f Γ± , regarded as operators from W p (D) to Lploc (Γ± ), are restrictions of the same operators acting from W 1 (D) to L1loc (Γ± ). 156

∞ (D) ⊂ W p (D) for all 1  p < ∞, the traces are also defined for Remark 2.2. Since W  f ∈ W ∞ (D); moreover, f Γ± ∈ L∞ (Γ± ) by Theorem 2.2 and Corollary 2.4.  It is known (cf., for example, [7]–[11]) that the traces f Γ± of a function f ∈ W p (Dj ) with j

p ± 1  p < ∞ do not necessarily belong to Lp (Γ± j ) and even to L (Γj ). As is shown in [7]–[9], the   trace operators f → f Γ+ and f → f Γ+ are linear continuous operators from W p (Dj ) to the j

j

p − (Γ+ ) and L p + (Γ− ), where τ − and τ + are defined in (1.2) and (1.3). weighted spaces L j j τ τ Let 1  p  ∞. We introduce the spaces  p (Dj ) = {f ∈ W p (Dj ) | f  − ∈ L p (Γ− )}, W Γj

j

 p (D) = {f ∈ W p (D) | f  − ∈ L p (Γ− )}. W Γ  p (Dj ) we have f  + ∈ L p (Γ+ ) and the Green formula It follows from [8, 11, 1] that for f ∈ W j Γ j

(“integration by parts”) holds:      + −.  ω · ∇f dωdx = f Γ+ dΓ − f Γ− dΓ j

Γ+ j

Dj

j

Γ− j

2.4. Additional information about the space W p (Dj ). We formulate a number of useful assertions, proved in [1], about properties of the space W p (Dj ). Lemma 2.1. Let f ∈ L1 (Dj ). Then  Dj

 Dj

+

   τ (ω,x) − (ω, x), f (ω, x) dω dx = f (ω, x + t ω) dt dΓ Γ− j

(2.2)

0 −

   τ (ω,x) + (ω, x). f (ω, x) dω dx = f (ω, x − t ω) dt dΓ Γ+ j

(2.3)

0

Theorem 2.1. A function w ∈ L1 (Dj ) is the weak directional derivative of f ∈ L1 (Dj ) in a direction ω (i.e., w = ω · ∇f ) if and only if the following property holds.   + 1,1 0, τ (ω, x) (P − ) For almost all (ω, x) ∈ Γ− j the function fω,x (t) = f (ω, x + t ω) belongs to W and   d fω,x (t) = wω,x (t) for a.a. t ∈ 0, τ + (ω, x) , dt where wω,x (t) = w(ω, x + t ω). Corollary 2.1. A function w ∈ L1 (Dj ) is the weak directional derivative of f ∈ L1 (Dj ) in a direction ω (i.e., w = ω · ∇f ) if and only if the following property holds.   − 1,1 0, τ (ω, x) (P + ) For almost all (ω, x) ∈ Γ+ j the function fω,x (t) = f (ω, x − t ω) belongs to W and   d fω,x (t) = −wω,x (t) for a.a. t ∈ 0, τ − (ω, x) , dt where wω,x (t) = w(ω, x − t ω). 157

Corollary 2.2. Let f ∈ W 1 (Dj ), and let fδ (ω, x) = δ

−1

δ f (ω, x + t ω) dt,

fδ (ω, x) = δ

−1

0

δ f (ω, x − t ω) dt. 0

Then the following assertions hold. 1. For almost all (ω, x) ∈ Γ− j the function fω,x (t) = f (ω, x + t ω) belongs to the space   + 1,1 W 0, τ (ω, x) . As a consequence, for almost all (ω, x) ∈ Γ− j there exist finite limits lim fδ (ω, x) = lim ap f (ω, x + tω),

δ→0+

+

t→0+

lim fδ (ω, X (ω, x)) = lim ap f (ω, x + tω).

δ→0+

t→ τ + (ω,x)

2. For almost all (ω, x) ∈ Γ+ j the function fω,x (t) = f (ω, x − t ω) belongs to the space   W 1,1 0, τ − (ω, x) . As a consequence, for almost all (ω, x) ∈ Γ+ j there exist finite limits lim fδ (ω, x) = lim ap f (ω, x − tω),

δ→0+

−

t→0+

lim fδ (ω, X (ω, x)) = lim ap f (ω, x − tω).

δ→0+

t→ τ − (ω,x)

Theorem 2.2. Let f ∈ W p (Dj ), 1  p  ∞. Then  f Γ− (ω, x) = lim fδ (ω, x) = lim ap f (ω, x + tω) δ→0+ t→0+  f Γ+ (ω, x) = lim fδ (ω, x) = lim ap f (ω, x − tω) δ→0+

for a.a. (ω, x) ∈ Γ− j , for a.a. (ω, x) ∈ Γ+ j .

t→0+

Corollary 2.3. Let f ∈ W 1 (Dj ). Then the following assertions hold. 1. For almost all (ω, x) ∈ Γ− + (ω, x)) j and almost all t ∈ (0, τ t

 f

Γ− j

(ω, x) = f (ω, x + t ω) −

ω · ∇f (ω, x + sω) ds.

(2.4)

0

2. For almost all (ω, x) ∈ Γ+ − (ω, x)) j and almost all t ∈ (0, τ  f

Γ+ j

t (ω, x) = f (ω, x − t ω) +

ω · ∇f (ω, x − sω) ds,

(2.5)

0

  ∞ −  Corollary 2.4. If f ∈ W ∞ (Dj ), then f Γ+ ∈ L∞ (Γ+ j ), f Γ− ∈ L (Γj ); moreover, j



f Γ+ L∞ (Γ+ )  f L∞ (Dj ) , j

j

j



f Γ− L∞ (Γ− )  f L∞ (Dj ) . j

j

  Corollary 2.5. If f ∈ W p (Dj ), 1  p  ∞ and f  0, then f Γ+  0 and f Γ−  0. j

158

j

 p (Dj ), 1  p < ∞. Then f  + ∈ L 1 (Dj ), p (Γ+ ), |f |p ∈ W Corollary 2.6. Let f ∈ W j Γ j

ω · ∇|f |p = p|f |p−1 sgn (f )ω · ∇f , and the Green formula (“integration by parts”) holds:     p +  p −.  ω · ∇|f | dωdx = |f Γ+ | dΓ − |f Γ− |p dΓ j

j

Γ+ j

Dj

(2.6)

Γ− j

1,p (Γ± ). Let 1 < p < ∞. We 2.5. Embedding of W p (Dj ) to the anisotropic spaces L j set  C1,p (Gj ) =

 ess sup x∈∂Gj

Ω− j (x)

 C2,p (Gj ) =

 ess sup x∈∂Gj

|ω · nj (x)| dω + τ (ω, x)1/(p−1)

1− 1

p

, 1− 1

p

τ + (ω, x)|ω · nj (x)| dω

.

Ω− j (x)

2.3. Let 1 < p < ∞, and let Gj be such that C1,p (Gj ) < ∞. If f ∈ W p (Dj ), then  Theorem ± 1,p (Γ ) and the following estimate holds: f Γ± ∈ L j j



f Γ± L1,p (Γ± )  C1,p (Gj ) f Lp (Dj ) + C2,p (Gj ) ω · ∇f Lp (Dj ) . j

j

(2.7)

Proof. From (2.4) it follows that for almost all (ω, x) ∈ Γ− j τ+(ω,x)

 |f 

1 (ω, x)|  + Γ− j τ (ω, x)

|f (ω, x + t ω)| dt + 0

⎡ 

1 1

τ + (ω, x) p

⎢ ⎣

+ τ (ω, x)

1− p1

|ω · ∇f (ω, x + t ω)| dt 0

⎤1

τ+(ω,x)

⎡ +

τ+(ω,x)

⎢ ⎣

p

⎥ |f (ω, x + t ω)|p dt⎦

0

⎤1

τ+(ω,x)

p

⎥ |ω · ∇f (ω, x + t ω)| dt⎦ , p

0

which implies in view of (2.2) 

f 

  

L1,p (Γ− ) Γ− j j

= ∂Gj

 

 ess sup x∈∂Gj



Ω− j (x)

 |f 

Γ− j

(ω, x)||ω · nj (x)| dω

p

1 dσ(x)

p

Ω− j (x)

|ω · nj (x)| dω τ + (ω, x)1/(p−1)

1− 1    p Γ− j

τ+(ω,x)

 1 p − (ω, x) |f (ω, x + t ω)|p dt dΓ

0

159





+ ess sup

+

1− 1    τ (ω,x)  1 p p + p − τ (ω, x)|ω · nj (x)| dω |ω · ∇f (ω, x + t ω)| dt dΓ (ω, x)

x∈∂Gj Ω− j (x)

0

Γ− j

= C1,p (Gj ) f Lp (Dj ) + C2,p (Gj ) ω · ∇f Lp (Dj ) .  The estimate (2.7) for f Γ− L1,p (Γ− ) is proved. j

j

Similarly, from (2.5) we derive the estimate  1 1 1,p (Gj )1− p f Lp (D ) + C 2,p (Gj )1− p ω · ∇f Lp (D ) ,

f Γ+ L1,p (Γ+ )  C j j j

j

where

 1,p (Gj ) = C

 ess sup x∈∂Gj

Ω+ j (x)

 2,p (Gj ) = C



1− 1

p

, 1− 1

p

−

τ (ω, x)ω · nj (x) dω

ess sup x∈∂Gj

ω · nj (x) dω τ − (ω, x)1/(p−1)

.

Ω+ j (x)

2,p (Gj ) = C2,p (Gj ) since τ − (ω, x) = τ + (−ω, x). 1,p (Gj ) = C1,p (Gj ) and C It remains to note that C Theorem is proved. We formulate a simple sufficient condition for the validity C1,p (Gj ) < ∞. Lemma 2.2. Let ∂Gj ∈ C 1+λ with some λ ∈ (0, 1). Then C1,p (Gj ) < ∞ for all p ∈ (1 + (2λ)−1 , ∞). Proof. Let (ω, x0 ) ∈ Γ− j . We set ω0 = nj (x0 ), μj = ω · nj (x0 ) and c0 = min{r0 , h0 }. It is + + (ω, x0 ) ∈ Cr ,−h ,h (ω0 , x0 ). Since easy to see that τ (ω, x0 )  c0 implies X 0

+

+

0

0

+

X (ω, x0 ) = x0 + τ (ω, x0 )ω0 = x0 + τ (ω, x0 )Pω0 ω + τ + (ω, x0 )μj ω0 , τ + (ω, x0 )Pω0 ω) and the inequality (1.1) where τ + (ω, x0 )Pω0 ω ∈ V r0 ,ω0 , we have τ + (ω, x0 )μj = γ( implies the estimates τ + (ω, x0 )Pω0 ω|1+λ  Aλ τ + (ω, x0 )1+λ . τ + (ω, x0 )|μj |  Aλ | Thus,

1/λ

1/λ

−1/λ + c−1 τ + (ω, x0 )−1  max{Aλ |μj |−1/λ , c−1 0 }  Aλ |μj | 0 .

Passing to the spherical coordinates, we find  1 1− 1 p  1/(p−1) 1/λ C1,p (Gj )  2π μ max{Aλ μ−1/λ , c−1 } dμ 0 0 1 p

1/(λp)

 1

 (2π) Aλ

1− 1 μ1−1/[λ(p−1)] dμ

p

1

− p1

+ π 1− p c0

< ∞.

0

The lemma is proved. Corollary 2.7. Let ∂Gj ∈ C 2 . Then C1,p (Gj ) < ∞ for all p ∈ (3/2, ∞). 160

3

Boundary Conditions

3.1. The laws of diffuse reflection and diffuse refraction. We recall some laws of diffuse reflection and refraction (cf. [12]). We denote by J(ω  , x) the intensity of the radiation propagating in the vacuum and falling on the surface at a point x in a direction ω  ∈ Ω− j (x). This radiation is partially diffusely reflected (i.e., it has the same intensity in all directions ω ∈ Ω+ j (x)) by the surface ∂Gj and partially diffusely refracted (i.e., it hasthe same intensity in all directions ω ∈ Ω− j (x)), entering the body Gj . The intensities of reflected and refracted radiations are independent of the propagation direction and are equal, respectively,  ρ+ j (x) J(ω  , x)|ω  · nj (x)| dω  , π Ω− j (x)

1 − ρ+ j (x) π



J(ω  , x)|ω  · nj (x)| dω  ,

Ω− j (x)

+ where ρ+ j (x) is the reflective ability of the surface ∂Gj at a point x. Note that 0 < ρj (x) < 1. + ∞ In addition, we assume that ρj ∈ L (∂Gj ) for all 1  j  m and

ρ+ = max ρ+ j L∞ (∂Gj ) < 1.

(3.1)

1jm

Let the radiation with intensity I(ω  , x) propagating in Gj fall on the surface ∂Gj at a point x in a direction ω  ∈ Ω+ j (x). This radiation is partially diffusely reflected (i.e., it has the same intensity in all directions ω ∈ Ω− j (x)) by ∂Gj and partially diffusely refracted (i.e., it has the same intensity in all directions ω ∈ Ω+ j (x)), going out from the body Gj to the vacuum. The intensities of the reflected and refracted radiations are given by the formulas  ρ− j (x) I(ω  , x)ω  · nj (x) dω  , π Ω+ j (x)

1 − ρ− j (x) π



I(ω  , x)ω  · nj (x) dω 

Ω+ j (x)

respectively. Here, ρ− j (x) = 1 −

1 (1 − ρ+ j (x)). kj2

− ∞ It is clear that 1 − 1/kj2 < ρ− j (x) < 1 and ρj ∈ L (∂Gj ) for all 1  j  m. We suppose that

ρ− = max ρ− j L∞ (∂Gj ) < 1.

(3.2)

1jm

3.2. Reflection and refraction operators. We set  1 ϕ(ω, x) ω · nj (x) dω, M + (ϕ)(x) = Mj+ (ϕ)(x) = π

x ∈ ∂Gj , 1  j  m,

Ω+ j (x)

161

1 π

M − (ψ)(x) = Mj− (ψ)(x) =

 ψ(ω, x) |ω · nj (x)| dω,

x ∈ ∂Gj , 1  j  m.

Ω− j (x)

older We note that Mj+ (1)(x) = 1, Mj− (1)(x) = 1, x ∈ ∂Gj . Hence for all 1  p < ∞, by the H¨ inequality, |Mj+ (ϕ)|p  Mj+ (|ϕ|p ) |Mj− (ψ)|p



Mj− (|ψ|p )

p (Γ+ ), ∀ϕ ∈ L j

(3.3)

p

(3.4)

∀ψ ∈ L

(Γ− j ).

− + We introduce the operators Rd,j and Rd,j of diffuse inner and outer reflection by the formulas − (ϕ)(ω, x) Rd,j

=

+ Rd,j (ψ)(ω, x)

=

+ ρ− j (x)Mj (ϕ)(x)

− ρ+ j (x)Mj (ψ)(x)

=

=



ρ− j (x) π

(ω, x) ∈ Γ− j ,

Ω+ j (x)



ρ+ j (x) π

ϕ(ω  , x)ω  · nj (x) dω  ,

ψ(ω  , x)|ω  · nj (x)| dω  ,

(ω, x) ∈ Γ+ j .

Ω− j (x)

We set − (ϕ)(ω, x), Rd− (ϕ)(ω, x) = Rd,j

(ω, x) ∈ Γ− j ,

1  j  m,

+ (ψ)(ω, x), Rd+ (ψ)(ω, x) = Rd,j

(ω, x) ∈ Γ+ j ,

1  j  m.

− + and Pd,j of diffuse refraction inside and outside Gj by We also introduce the operators Pd,j the formulas − − (ψ)(ω, x) = (1 − ρ+ Pd,j j (x))Mj (ψ)(x)

=



1 − ρ+ j (x) π

ψ(ω  , x)|ω  · nj (x)| dω  ,

(ω, x) ∈ Γ− j ,

(3.5)

(ω, x) ∈ Γ+ j .

(3.6)

Ω− j (x)

and + + (ϕ)(ω, x) = (1 − ρ− Pd,j j (x))Mj (ϕ)(x)

=

1 − ρ− j (x) π



ϕ(ω  , x) ω  · nj (x) dω  ,

Ω+ j (x)

We set − (ψ)(ω, x), Pd− (ψ)(ω, x) = Pd,j

(ω, x) ∈ Γ− j ,

1  j  m,

+ Pd+ (ϕ)(ω, x) = Pd,j (ϕ)(ω, x),

(ω, x) ∈ Γ+ j ,

1  j  m.

3.3. Preliminary formulation of the reflection–refraction conditions. We formulate conditions satisfied by the radiation intensity on different parts of the boundary. For the 162

radiation propagating in G we denote by I|Γ± and I|Γ± the values of intensity on Γ± and Γ± j j

respectively. For the radiation propagating in the vacuum the values of I on Γ− are denoted by J. For (ω, x) ∈ S − the radiation entering into G is composed of the diffusely reflected and diffusely refracted radiations: I|Γ− = Rd− (I|Γ+ ) + Pd− (J),

(ω, x) ∈ S − .

(3.7)

∗

For (ω, x) ∈ S − the radiation J falling on ∂G from the vacuum goes from outside, and we can assume that it is prescribed: J = J∗ ,

∗

(ω, x) ∈ S − .

(3.8)

For (ω, x) ∈ S− the radiation J falling on ∂G from the vacuum comes from a point ∈ ∂G. It is composed of the diffusely reflected and diffusely refracted radiations − at X (ω, x):

X − (ω, x)

J(ω, x) = Rd+ (J)(ω, X − (ω, x)) + Pd+ (I|Γ+ )(ω, X − (ω, x)),

(ω, x) ∈ S− .

(3.9)

We introduce the translation operator T by the formula T ϕ(ω, x) = ϕ(ω, X − (ω, x)),

(ω, x) ∈ S−

and write the condition (3.9) in the form J = T Rd+ (J) + T Pd+ (I|Γ+ ),

(ω, x) ∈ S− .

(3.10)

3.4. The reflection–refraction conditions at points of tangency of the surfaces ∂Gi and ∂Gj . Let x be a point of tangency of ∂Gi and ∂Gj for some i = j. We recall that Gi and Gj are separated by the vacuum and, consequently, they are separated by an infinitely thin − − + vacuum layer at tangency points. We note that nj (x) = −ni (x) and set Γ− ij = Γi ∩ Γj = Γi ∩ Γj + + − and Γ+ ij = Γi ∩ Γj = Γi ∩ Γj . Let Ji be the intensity of the radiation propagating in the vacuum layer between the surfaces ∂Gi and ∂Gj in a direction ω ∈ Ω− i (x) and coming to a point x ∈ ∂Gi from the side of ∂Gj , whereas Jj is the intensity of the similar radiation coming to x from the side of ∂Gi . They are composed of diffusely reflected and refracted radiations. Therefore, they are independent of ω and satisfy the equations − + Ji = ρ+ j Jj + (1 − ρj )Mj (I|Γ+ ),

(3.11)

− + Jj = ρ+ i Ji + (1 − ρi )Mi (I|Γ+ ).

(3.12)

j

i

From (3.11) and (3.12) we easily find Ji =

  + 1 − + ) + (1 − ρ− )Mj+ (I|Γ+ ) . j + + ρj (1 − ρi )Mi (I|Γ+ i j 1 − ρi ρj

(3.13)

163

Let I|Γ− be the intensity of the radiation entering into Gi in a direction ω ∈ Ω− i (x) at a point i x of tangency of the surfaces ∂Gi and ∂Gj . This radiation is composed of diffusely reflected and refracted radiations. Therefore, I|Γ− is independent of ω and satisfies the equality i

I|Γ− = i

+ ρ− ) i Mi (I|Γ+ i

+ (1 − ρ+ i )Ji ,

(ω, x) ∈ Γ− ij .

Substituting (3.13) in this equality, we find + − + I|Γ− = ρ− ij Mi (I|Γ+ ) + (1 − ρji )Mj (I|Γ+ ), i

i

j

(ω, x) ∈ Γ− ij ,

(3.14)

where ρ− ij = 1 −

+ (1 − ρ− i )(1 − ρj ) + 1 − ρ+ i ρj

− 1 − ρ− ji = (1 − ρij )

,

− (1 − ρ+ ki2 i )(1 − ρj ) = . + kj2 1 − ρ+ i ρj

+ + − + + ∞ ∞ Remark 3.1. From ρ− i , ρi ∈ L (∂Gi ), ρj ∈ L (∂Gj ), 0 < ρi < 1, 0 < ρi < 1, 0 < ρj < 1 − − − it follows that ρij ∈ L∞ (∂Gi ∩ ∂Gj ), 0 < ρi < ρij < 1. − − and Pd,ij by We introduce the operators Rd,ij − + Rd,ij (ϕ)(ω, x) = ρ− ij (x)Mi (ϕ)(x), − Pd,ij (ψ)(ω, x)

= (1 −

(ω, x) ∈ Γ− ij ,

+ ρ− ji (x))Mj (ψ)(x),

(3.15)

(ω, x) ∈

Γ− ij .

(3.16)

Using these operators, we can write (3.14) in the form − − (I|Γ+ ) + Pd,ij (I|Γ+ ), I|Γ− = Rd,ij i

i

j

(ω, x) ∈ Γ− ij .

(3.17)

Remark 3.2. The equality (3.17) remains valid if Gi and Gj are not separated by an infinitely thin vacuum layer at the points of tangency of ∂Gi and ∂Gj , but there is an ideal − optical contact. This case differs from the above one only by other values of ρ− ij (x) and ρji (x) in the definitions (3.15) and (3.16) of the diffuse reflection and refraction operators. However, − − − 2 2 as above, 0 < ρ− ij < 1 and 0 < ρji < 1; moreover, 1 − ρji = (1 − ρij )(ki /kj ). 3.5. Preliminary statement of the boundary value problem. At the physical level, the problem can be stated as follows: find a function I defined on D = Ω × G and a function J defined on S − that satisfy the equation ω · ∇I + βI = sS (I) + κk 2 F,

(ω, x) ∈ D

(3.18)

and the conditions − − I|Γ− = Rd,ij (I|Γ+ ) + Pd,ij (I|Γ+ ), i

j

(ω, x) ∈ Γ− ij ,

i = j,

(3.19)

I|Γ− = Rd− (I|Γ+ ) + Pd− (J),

(ω, x) ∈ S − ,

(3.20)

J = T Rd+ (J) + T Pd+ (I|Γ+ ),

(ω, x) ∈ S− ,

(3.21)

J = J∗ , 164

i

∗

(ω, x) ∈ S − .

(3.22)

In Subsection 4.5, we exclude the function J from the problem by determining it from (3.21) and (3.22). In Section 5, we give a final setting of the problem and a rigorous definition of its solution.

4

Properties of the Scattering and Boundary Operators

Hereinafter, p, q ∈ [1, ∞], 1/p + 1/q = 1, i.e., p and q are the H¨ older conjugate exponents. For the sake of brevity we set k(ω, x) = kj for (ω, x) ∈ Dj \ Σj , 1  j  m, and define the ± ∞ functions ρ± (x) = ρ± j (x), x ∈ ∂Gj \ Σj , 1  j  m, on ∂G \ Σ. We recall that ρj ∈ L (∂Gj ), ± + − 2 0 < ρj < 1, 1 − ρj = (1 − ρj )kj , 1  j  m. Furthermore, ess sup ρ+ (x)  ρ+ < 1,

(4.1)

x∈∂G\Σ

ess sup ρ− (x)  ρ− < 1.

(4.2)

x∈∂G\Σ

4.1. The scattering operator S . We recall that the scattering operator S is defined by  1 S (I)(ω, x) = θj (ω · ω  )I(ω  , x) dω  , (ω, x) ∈ Dj , 1  j  m, 4π Ω

with the indicatrix θj such that θj  0, θj ∈ L1 (−1, 1), and 1 2

1 θj (μ) dμ = 1. −1

As is known, S : Lp (D) → Lp (D), 1  p  ∞, and S (I) Lp (D)  I Lp (D) . We also pay attention to the following simple formulas:   S (I) dω = I dω ∀ I ∈ L1 (D), S (1) = 1, Ω

Ω

and the inequality |S (I)|p  S (|I|p )

∀ I ∈ Lp (D),

1  p < ∞.

− + − + 4.2. The operators Rd,j , Rd,j , Pd,j , and Pd,j . The following assertions are obvious.

− + Lemma 4.1. 1. For all 1  p  ∞ the operators Rd,j and Rd,j are linear bounded operators + − − + p (Γ ) and from L 1,p (Γ ) to L p (Γ ) respectively. 1,p (Γ ) to L from L j j j j − + 1,p (Γ− ) 2. For all 1  p  ∞ the operators P and P are linear bounded operators from L d,j

j

d,j

p (Γ− ) and from L 1,p (Γ+ ) to L p (Γ+ ) respectively. to L j j j

1,p (Γ− ), 1  j  m. Then 1,p (Γ+ ), ψ ∈ L Lemma 4.2. Suppose that 1  p < ∞, ϕ ∈ L j j −2/q

− (ϕ) + kj |Rd,j

2/q

− + + − p p Pd,j (ψ)|p  ρ− j [Mj (|ϕ|)] + (1 − ρj )[Mj (|ψ|)] ,

+ + − − + p p (ψ) + kj Pd,j (ϕ)|p  ρ+ |Rd,j j [Mj (|ψ|)] + (1 − ρj )[Mj (|ϕ|)] ,

(ω, x) ∈ Γ− j , (ω, x) ∈ Γ+ j .

(4.3) (4.4) 165

Proof. Taking into account the convexity of f (y) = |y|p , from the equalities 1 − ρ+ j = − 2 kj (1 − ρj ) and the inequalities (3.3), (3.4) we find −2/q

− (ϕ) + kj |Rd,j

2/p

− + − − p Pd,j (ψ)|p = |ρ− j Mj (ϕ) + (1 − ρj )kj Mj (ψ)|

+ − 2 − − + + − p p p p  ρ− j |Mj (ϕ)| + (1 − ρj )kj |Mj (ψ)|  ρj [Mj (|ϕ|)] + (1 − ρj )[Mj (|ψ|)] .

Similarly, 2/q

−2/p

+ + − + (ψ) + kj Pd,j (ϕ)|p = |ρ+ |Rd,j j Mj (ψ) + (1 − ρj )kj

Mj+ (ϕ)|p

− + −2 + + − − + p p p p  ρ+ j |Mj (ψ)| + (1 − ρj )kj |Mj (ϕ)|  ρj [Mj (|ψ|)] + (1 − ρj )[Mj (|ϕ|)] .

The lemma is proved. − − 4.3. The operators Rd,ij and Pd,ij . Let Γ− ij = ∅ for some i = j. − − and Pd,ij are linear bounded operators Lemma 4.3. For all 1  p  ∞ the operators Rd,ij + − + − 1,p p 1,p p (Γ ) and from L (Γ ) to L (Γ ) respectively. (Γ ) to L from L ij

ij

ji

ij

1,p (Γ+ ). Then 1,p (Γ+ ), and ψ ∈ L Lemma 4.4. Suppose that 1  p < ∞, ϕ ∈ L ij ji − − + − + p p (ϕ) + (ki /kj )−2/q Pd,ij (ψ)|p  ρ− |Rd,ij ij [Mi (|ϕ|)] + (1 − ρji )[Mj (|ψ|)] ,

(ω, x) ∈ Γ− ij . (4.5)

Proof. As in the proof of the estimate (4.3), we have − − + − 2/p (ϕ) + (ki /kj )−2/q Pd,ij (ψ)|p = |ρ− Mj+ (ψ)|p |Rd,ij ij Mi (ϕ) + (1 − ρij )(ki /kj ) + − + − + − + p 2 p p p  ρ− ij |Mi (ϕ)| + (1 − ρij )(ki /kj ) |[Mj (ψ)| = ρij x[Mi (|ϕ|)] + (1 − ρji )[Mj (|ψ|)] .



Corollary 4.1. For 1  p < ∞ the following inequality holds: − − (ϕ) + (ki /kj )−2/q Pd,ij (ψ) pp

Rd,ij

L (Γ− ij )

− − + Rd,ji (ψ) + (kj /ki )−2/q Pd,ji (ϕ) pp

L (Γ− ji )

 ϕ pp

L (Γ+ ij )

+ ψ pp

L (Γ+ ji )

(4.6)

p (Γ+ ). p (Γ+ ) and ψ ∈ L for all ϕ ∈ L ij ji Proof. From the inequalities (4.5) and (3.3), (3.4) it follows that − − − − (ϕ) + (ki /kj )−2/q Pd,ij (ψ) pp − + Rd,ji (ψ) + (kj /ki )−2/q Pd,ji (ϕ) pp −

Rd,ij L (Γij ) L (Γji )    + + + p − p − p −  ρ− (1 − ρ− ρ− ij Mi (|ϕ| )dΓ + ji )Mj (|ψ| )dΓ + ji Mj (|ψ| )dΓ Γ− ij

Γ− ij



(1 −

+

+ p − ρ− ij )Mi (|ϕ| )dΓ

=π ∂Gi ∩∂Gj

Γ− ji

= ϕ pp

L (Γ+ ij )

166



+ ψ pp

L (Γ+ ji )

.

Γ− ji

Mi+ (|ϕ|p ) dσ



+π

Mj+ (|ψ|p ) dσ

∂Gi ∩∂Gj



4.4. The operator T . We recall (cf. Subsection 3.3) that the operator T is defined by T ϕ(ω, x) = ϕ(ω, X − (ω, x)),

(ω, x) ∈ S− .

The following assertion was proved in [1]. p (S− ), 1  p  Lemma 4.5. The operator T is a linear bounded operator from Lp (S+ ) to L ∞, and   − + ∀ϕ ∈ L 1 (S+ ). T ϕ dΓ = ϕ dΓ (4.7) − S

+ S

Corollary 4.2. The following equality holds:

T (|ϕ|) Lp (S− ) = ϕ Lp (S+ )

p (S+ ), ∀ϕ ∈ L

1  p  ∞.

(4.8)

4.5. The operators Bd and Cd . We recall that the function J in (3.18)–(3.22) satisfies J = T Rd+ (J) + T Pd+ (I|Γ+ ), J = J∗ ,

(ω, x) ∈ S− ,

(4.9)

∗

(ω, x) ∈ S − .

∗

(4.10)

◦

◦

We also recall that S − = S− ∪ S − ∪ S − , where, as was proved in [1], meas (S − ; dΓ) = 0. ∗

1,p − (S + ), 1  p  ∞. By a solution to the problem (4.9), (4.10) Let J∗ ∈ L1,p (S + ), I|Γ+ ∈ L 1−ρ ∗

1,p (S − ) satisfying Equation (4.9) a.e. on S− and equal to J∗ on S − . we mean a function J ∈ L We show that J can be excluded from the problem by solving the system (4.9), (4.10). Let χS− and χS+ be the characteristic functions of the sets S− and S+ respectively. Let  1 − α(x) = M (χS− )(x) = χS− (ω, x)|ω · n(x)| dω, x ∈ ∂G \ Σ. (4.11) π Ω− i (x)

We note that 0  α  1, α ∈ L∞ (∂G \ Σ). Furthermore, χS− (ω, x) = χS+ (−ω, x) implies  1 χS+ (ω, x) ω · n(x) dω, x ∈ ∂G \ Σ. α(x) = M + (χS+ )(x) = π Ω+ i (x)

1,p (S− ) → L 1,p (S + ) by We introduce the operator Rd+ : L Rd+ (ψ)(ω, x) = Rd+ (χS− ψ)(ω, x),

(ω, x) ∈ S + .

We set J∗ = 0 on S− and reduce the system (4.9), (4.10) to the equation J = T Rd+ (J) + T Rd+ (J∗ ) + T Pd+ (I|Γ+ ),

(ω, x) ∈ S− .

(4.12)

Lemma 4.6. For all 1  p  ∞ 1

T Rd+ L1,p (S− )→L1,p (S− )  α p ρ+ L∞ (∂G\Σ)  ρ+ < 1. 167

p (S− ), 1  p < ∞. By (4.8) and (4.1), we have Proof. Let ϕ ∈ L 1 π p−1

T Rd+ (ϕ) p1,p L

− ) (S

 T Rd+ (|ϕ|) pp

− ) L (S

= Rd+ (|ϕ|) pp

+ ) L (S

= (ρ+ )p [M − (χS− |ϕ|)]p L1 (S+ ) = πα(ρ+ )p [M − (χS− |ϕ|)]p L1 (∂G\Σ) =

1 π p−1

1

α p ρ+ ϕ p1,p L

− ) (S



1

1

π p−1

α p ρ+ pL∞ (∂G\Σ) ϕ p1,p L

− ) (S

1



π p−1

(ρ+ )p ϕ p 1,p L

− ) (S

.

∞ (S− ), then the above estimate implies If p = ∞ and ϕ ∈ L

T Rd+ (ϕ) L1,p (S− )  ρ+ ϕ L1,p (S− ) . Passing to the limit as p → ∞, we see that

T Rd+ (ϕ) L 1,∞ (S− )  ρ+ ϕ L 1,∞ (S− ) .



1,p (S− ) → L 1,p (S− ), Corollary 4.3. There exists a linear bounded operator (Ip −T Rd+ )−1 : L 1,p − (S ). where Ip is the identity operator in L Corollary 4.4. A solution J to the problem (4.9), (4.10) exists, is unique, and J = Bd (I|Γ+ ) + Cd (J∗ ), where Bd (I|Γ+ )(ω, x) =

Cd (J∗ )(ω, x) =

(4.13)

⎧ ⎨(Ip − T Rd+ )−1 T Pd+ (I|Γ+ )(ω, x),

(ω, x) ∈ S− ,

⎩0,

(ω, x) ∈ S − ,

∗

⎧ ⎨(Ip − T Rd+ )−1 T Rd+ (J∗ )(ω, x),

(ω, x) ∈ S− ,

⎩J (ω, x), ∗

(ω, x) ∈ S − .

∗

∗

1,p (S − ), I|Γ+ ∈ L 1,p − (S + ), 1  p  ∞. Then a solution Theorem 4.1. Suppose that J∗ ∈ L 1−ρ J to the problem (4.9), (4.10) exists, is unique, and satisfies the estimate

J p1,p L

(S 1−αρ+

−)

 J∗ p

∗

 1,p (S − ) L

+ k −2/q I|Γ+ p1,p L

or

J L1,∞ (S − )  max{ J∗ 1,∞ L

∗

(S − )

α(1−ρ− )

(S + )

,

, k −2 I|Γ+ L1,∞ (S + ) },

1  p < ∞, p = ∞.

(4.14) (4.15)

Proof. The existence and uniqueness of a solution J is established in Corollary 4.4. Let us obtain the estimate (4.14). Taking into account (4.8) and (4.4), from (4.9) we find 1

J p1,p −  J pp −  T (Rd+ (|J|) + Pd+ (|I|Γ+ |)) pp − L (S ) L (S ) L (S ) π p−1 = Rd+ (|J|) + Pd+ (|I|Γ+ |) pp

+ ) L (S

= Rd+ (|J|) + k 2/q Pd+ (k −2/q |I|Γ+ |) pp

+ ) L (S

 ρ+ [M − (|J|)]p + (1 − ρ− )[M + (k −2/q |I|Γ+ |)]p L1 (S+ ) = παρ+ [M − (|J|)]p L1 (∂G\Σ) + πα(1 − ρ− )[M + (k −2/q |I|Γ+ |)]p L1 (∂G\Σ) =

168

1 1

J p1,p − + p−1 k −2/q I|Γ+ p1,p . (S + ) L + (S ) L π p−1 π αρ α(1−ρ− )

As a consequence,

J p1,p L

p

Adding J∗

 1,p L

 J∗ p 1,p  L

αρ

∗

− + (S )

+ k −2/q I|Γ+ p1,p L

α(1−ρ− )

(S + )

.

to the left-hand and right-hand sides, we obtain (4.14).

∗

1−αρ+

− )

(S 1−αρ+

(S − )

The estimate (4.15) is obtained from the estimate (4.14) written as

J L1,p

1−αρ

− + (S )

  J∗ p

∗

 1,p (S − ) L

1

+ k −2/q I|Γ+ p1,p L

α(1−ρ− )

(S + )

p

by the limit passage as p → ∞. Remark 4.1. Since the solution to the problem (4.9), (4.10) satisfies (4.13), the estimates (4.14) and (4.15) can be written as

Bd (I|Γ+ ) + Cd (J∗ ) p1,p L

1−αρ

 J∗ p

− + (S )

∗

 1,p (S − ) L

Bd (I|Γ+ ) + Cd (J∗ ) L 1,∞ (S − )  max{ J∗  1,∞ L

+ k −2/q I|Γ+ p1,p L

∗

(S − )

α(1−ρ− )

(S + )

, 1  p < ∞, (4.16)

, k −2 I|Γ+ L 1,∞ (S + ) }.

(4.17)

We also note that I|Γ+  0 implies Bd (I|Γ+ )  0 and J∗  0 implies Cd (J∗ )  0. The following lemma provides a simple, but important property of the operators Bd and Cd . ∗  Lemma 4.7. Suppose that I Γ+ = k 2 M on Γ+ and J ∗ = M on S − , where M = const. Then

Bd (I|Γ+ ) + Cd (J ∗ ) = M

a.e. on S − .

(4.18)

Proof. We set J = M on S − and show that J is a solution to the problem (4.9), (4.10) corresponding to J∗ = J ∗ . Indeed, for (ω, x) ∈ S− we have T Rd+ (J)+T Pd+ (I|Γ+ ) = T (Rd+ (M )+ ∗

Pd+ (k 2 M )) = T (ρ+ M + (1 − ρ+ )M ) = T (M ) = M = J. It is obvious that J = J ∗ on S − .

5 Boundary Value Problem for the Radiation Transfer Equation with Diffuse Reflection and Refraction Conditions 5.1. Statement of the problem. We study the boundary value problem ω · ∇I + βI = sS (I) + κk 2 F,

(ω, x) ∈ D,

− − I|Γ− = Rd,ij (I|Γ+ ) + Pd,ij (I|Γ+ ), i

i

j

(ω, x) ∈ Γ− ij ,

I|Γ− = Rd− (I|Γ+ ) + Pd− Bd (I|Γ+ ) + Pd− Cd (J∗ ),

(5.1) i = j, (ω, x) ∈ S − ,

(5.2) (5.3)

obtained from the problem (3.18)–(3.22) by eliminating J = Bd (I|Γ+ ) + Cd (J∗ ). Assume that ∗ 1,p (S − ), 1  p  ∞. F ∈ Lp (D), J∗ ∈ L p (D) that satisfies By a solution to the problem (5.1)–(5.3) we mean a function I ∈ W Equation (5.1) almost everywhere in D and the conditions (5.2) and (5.3) almost everywhere on − − Γ− ij and S respectively. If Γij are empty, the boundary conditions (5.2) are purely formal. 169

We recall that β = κ + s and, in each domain Gj , the absorption coefficient κ and the scattering coefficient s take constants values equal to κj > 0 and sj  0 respectively. We introduce the albedo coefficient  = s/(κ + s) with the values j = sj /(κj + sj ) on Gj , 1  j  m. We note that max = max1jm j < 1. We recall that k(ω, x) = kj for (ω, x) ∈ Dj \Σj , 1  j  m. 5.2. Auxiliary problem. We consider the auxiliary problem ω · ∇I + βI = f, I|Γ− = g,

(ω, x) ∈ Dj ,

(5.4)

(ω, x) ∈ Γ− j ,

(5.5)

with f ∈ L∞ (Dj ) and g ∈ L∞ (Γ− j ). ∞ A function I ∈ W (Dj ) is called a solution to the problem (5.4), (5.5) if it satisfies Equation (5.4) almost everywhere on Dj and the condition (5.5) almost everywhere on Γ− j . Regarding the following assertion we refer, for example, to [8, 11]. Theorem 5.1. A solution to the problem (5.4), (5.5) exists, is unique, and is represented as I(ω, x) = e−βj

τ− (ω,x)

− (ω, x)) + g(ω, X

τ−(ω,x)

e−βj t f (ω, x − t ω) dt,

(ω, x) ∈ Dj ,

(5.6)

0

− (ω, x) are defined in (1.3). where τ − (ω, x) and X By (5.6), the following assertion is obvious. Corollary 5.1. Let I be a solution to the problem (5.4), (5.5). 1. If f  0 and g  0, then I  0. 2. If f  βj Mj on Dj and g  Mj on Γ− j , where Mj = const, then I  Mj on Dj . 5.3. A priori estimates and uniqueness of a solution. We use the notation F+ = max{F, 0},

J∗+ = max{J∗ , 0},

F− = min{F, 0},

J∗− = min{J∗ , 0},

I+ = max{I, 0}, I− = min{I, 0}.

For p ∈ [1, ∞] we denote by q ∈ [1, ∞] the H¨ older conjugate exponent, i.e., 1/p + 1/q = 1. p (D) is a solution to the problem Theorem 5.2. Suppose that 1  p  ∞ and I ∈ W (5.1)–(5.3). Then for 1  p < ∞ the following estimates hold: ! !p 1

k −2/q I+ |Γ+ pp + k −2/q I+ pLp (D)  !F+ !Lp (D) + p−1 J∗+ p ∗ , (5.7) + L κ (S )  1,p (S − ) 2 π κk L (1−α)(1−ρ− )

k −2/q I− |Γ+ pp L

(1−α)(1−ρ− )

(S + )

! !p + k −2/q I− pLp (D)  !F− !Lp κ

κk

2 (D)

+

1 π p−1

J∗− p

∗

 1,p (S − ) L

(5.8)

and

k −2/q I|Γ+ pp L

(1−α)(1−ρ− )

κ −1/q k −2/q ω · ∇I pLp (D) 170

! !p + k −2/q I pLp (D)  !F !Lp

1

J∗ p ∗ ,  1,p (S − ) π p−1 L  p   2 1 p p 

F L 2 (D) + p−1 J∗

, ∗  1,p (S − ) κk 1 − max π L

(S + )

κ

(D) κk2

+

(5.9) (5.10)

whereas for p = ∞ the following estimates hold: " # 1

k −2 I+ L∞ (D)  max F+ L∞ (D) , J∗+ 1,∞ ∗ − , L (S ) π $ % 1

k −2 I− L∞ (D)  max F− L∞ (D) , J∗− 1,∞ ∗ − , L (S ) π and

" # 1

k −2 I L∞ (D)  max F L∞ (D) , J∗ 1,∞ ∗ − , L (S ) π # " 2 1

κ −1 k −2 ω · ∇I L∞ (D)  max F L∞ (D) , J∗ 1,∞ ∗ − . L (S ) 1 − max π

(5.11) (5.12)

(5.13) (5.14)

Proof. We begin with the estimate (5.7). We set Ip = k 2/p−2 I = k −2/q I and write Equation (5.1) in the form (5.15) ω · ∇Ip + βIp = sS (Ip ) + κk 2/p F. p (D) and ω · ∇I p = pI p−1 sgn (I+ ) ω · ∇Ip almost We set Ip,+ = max{Ip , 0}. Note that Ip,+ ∈ W p,+ p,+ p−1 everywhere on D. Multiplying Equation (5.15) by p Ip,+ and using the Young inequality, we find p p p−1 + pβIp,+ = p βIp,+ sgn (I+ ) (S (Ip ) + (1 − )k 2/p F ) ω · ∇Ip,+ p−1 sgn (I+ ) (S (Ip,+ ) + (1 − )k 2/p F+ )  p βIp,+ p  (p − 1) βIp,+ + β(S (Ip,+ ) + (1 − )k 2/p F+ )p p p  (p − 1) βIp,+ + β(S (Ip,+ ) + (1 − )k 2 F+p ).

Thus,

p p p + βIp,+  sS (Ip,+ ) + κk 2 F+p . ω · ∇Ip,+

Integrating over D and using (2.6), we find ! p

Ip,+ |Γ+ pp + + βIp,+

L1 (D)  !Ip,+ |Γ− pp L (Γ )

L (Γ− )

p + sS (Ip,+ ) L1 (D) + κk 2 F+p L1 (D)

L (Γ− )

p + sIp,+

L1 (D) + κk 2 F+p L1 (D) .

! = !Ip,+ |Γ− pp

Thus,

Ip,+ |Γ+ pp L

(Γ+ )

! p + κIp,+

L1 (D)  !Ip,+ |Γ− pp L

(Γ− )

! ! + !κk 2 F+p !L1 (D) .

(5.16)

− Since the condition (5.2) is satisfied almost everywhere on Γ− ij and Γji , we have − − Ip |Γ− = Rd,ij (Ip |Γ+ ) + (kj /ki )2/q Pd,ij (Ip |Γ+ )

a.e. on Γ− ij ,

− − (Ip |Γ+ ) + (ki /kj )2/q Pd,ji (Ip |Γ+ ) Ip |Γ− = Rd,ji

a.e. on Γ− ji .

i

j

i

j

j

i

Hence − − (Ip,+ |Γ+ ) + (kj /ki )2/q Pd,ij (Ip,+ |Γ+ ) Ip,+ |Γ−  Rd,ij

a.e. on Γ− ij ,

− − (Ip,+ |Γ+ ) + (ki /kj )2/q Pd,ji (Ip,+ |Γ+ ) Ip,+ |Γ−  Rd,ji

a.e. on Γ− ji .

i

j

i

j

j

i

171

Thus,

Ip,+ |Γ− pp

− −  Rd,ij (Ip,+ |Γ+ ) + (kj /ki )2/q Pd,ij (Ip,+ |Γ+ ) pp

,

Ip,+ |Γ− pp

− −  Rd,ji (Ip,+ |Γ+ ) + (ki /kj )2/q Pd,ji (Ip,+ |Γ+ ) pp

.

L (Γ− ij )

i

L (Γ− ji )

j

i

L (Γ− ij )

j

j

L (Γ− ji )

i

Adding these inequalities and using the estimate (4.6), we get

Ip,+ |Γ− pp i

L (Γ− ij )

Thus,

+ Ip,+ |Γ− pp

L (Γ− ji )

j

m  i,j=1,i=j

Ip,+ |Γ− pp − L (Γ ) i ij

 Ip,+ |Γ+ pp

L (Γ+ ij )

i

m 



i,j=1,i=j

+ Ip,+ |Γ+ pp j

Ip,+ |Γ+ pp i

L (Γ+ ij )

L (Γ+ ji )

.

.

(5.17)

Since the condition (5.3) is satisfied almost everywhere on S − , we have Ip,+ |Γ−  Rd− (Ip,+ |Γ+ ) + k −2/q Pd− (Bd (I+ |Γ+ ) + Cd (J∗+ ))

a.e. on S − .

By the inequality (4.3), p p Ip,+ |Γ−  ρ− M + (Ip,+ |Γ+ ) + (1 − ρ+ )[M − (Bd (I+ |Γ+ ) + Cd (J∗+ ))]p

a.e. on S − .

Using the estimate (4.16), we find

Ip,+ |Γ− pp

L (S − )

p  ρ− Ip,+ |Γ+ L1 (S + ) +

p  ρ− Ip,+ |Γ+ L1 (S + ) +

 Ip,+ |Γ+ pp L

ρ− +α(1−ρ− )

1 π p−1

(S + )

1

Bd (I+ |Γ+ ) + Cd (J∗+ ) p1,p L (S − ) π p−1 1−ρ+

Ip,+ |Γ+ p1,p

+

L

1 π p−1

α(1−ρ− )

J∗+ p

(S + )

∗

 1,p (S − ) L

+

1 π p−1

J∗+ p

∗

 1,p (S − ) L

.

(5.18)

Adding the inequalities (5.16), (5.17), and (5.18), we obtain the estimate

Ip,+ |Γ+ pp L

(1−α)(1−ρ− )

(S + )

! ! p + κIp,+

L1 (D)  !κk 2 F+p !L1 (D) +

1 π p−1

J∗+ p

∗

 1,p (S − ) L

,

i.e., the estimate (5.7). Taking into account that the problem under consideration is linear, we obtain the estimate (5.8) from (5.7). Combining (5.7) and (5.8), we arrive at the estimate (5.9). Then (5.15) implies the estimate (5.10) (cf. [1] for a detailed derivation of (5.10) from (5.15)). ∗

∞ (D) = W ∞ (D), then (5.11)–(5.14) are obtained 1,∞ (S − ) and I ∈ W If F ∈ L∞ (D), J∗ ∈ L from the estimates (5.7)–(5.10) raised to the 1/pth power by the limit passage as p → ∞. Corollary 5.2. Let I be 1) if F  0, J∗  0, then 2) if F  0, J∗  0, then 3) if F = 0, J∗ = 0, then

a solution to the problem (5.1)–(5.3). Then I  0, I  0, I = 0.

Corollary 5.3. If the problem (5.1)–(5.3) has a solution, then this solution is unique. 172

5.4. The existence of a solution. We prove that there exists a solution to the problem ∗ 1,∞ (S − ). (5.1)–(5.3). We begin with the case F ∈ L∞ (D), J∗ ∈ L ∗

1,∞ (S − ). Then the problem (5.1)–(5.3) has a unique Theorem 5.3. Let F ∈ L∞ (D), J∗ ∈ L solution I ∈ W ∞ (D). Proof. We begin with the case F  0, J∗  0. We set M = max{ F L∞ (D) , J∗  1,∞ ∗

L

∗

(S − )

}

and introduce the functions I = k 2 M , F = M on D and the function J ∗ = M on S − . It is easy to verify that ω · ∇I + βI = sS (I) + κk 2 F , I|Γ− = i

− Rd,ij (I|Γ+ ) i

+

(x, ω) ∈ D,

− Pd,ij (I|Γ+ ), j

(ω, x) ∈

(5.19) Γ− ij ,

I|Γ− = Rd− (I|Γ+ ) + Pd− Bd (I|Γ+ ) + Pd− Cd (J ∗ ),

i = j,

(5.20)

(ω, x) ∈ S − .

(5.21)

To check the equality (5.21), one should to use (4.18). Thus, the function I is a solution to the problem (5.1)–(5.3) with F = F , J∗ = J ∗ and simultaneously a solution to the problem (5.4), (5.5) with f = f = βj kj2 M on Dj , g = g = kj2 M on Γ− j , 1  j  m. We set I0 = 0 on D and consider the iteration process ω · ∇In+1 + βIn+1 = fn , In+1 |Γ− = gn ,

(x, ω) ∈ D,

(5.22)

−

(ω, x) ∈ Γ ,

(5.23)

where n  0 and fn = sS (In ) + κk 2 F, ⎧ − − (In |Γ+ ), ⎨Rd,ij (In |Γ+ ) + Pd,ij i j gn = ⎩ − Rd (In |Γ+ ) + Pd− Bd (In |Γ+ ) + Pd− Cd (J∗ ),

(ω, x) ∈ Γ− ij , i = j, (ω, x) ∈ S − .

We note that In ∈ W ∞ (D), 0  In  I implies fn ∈ L∞ (D), gn ∈ L∞ (Γ− ); moreover, 0  fn  f , 0  gn  g. Therefore, the existence of a solution In+1 ∈ W ∞ (D) to the problem (5.22), (5.23) for all n  0 follows from Theorem 5.1 and, by Corollary 5.1, 0  In  In+1  I for all n  0. Thus, there exists a function I ∈ L∞ (D) such that 0  I  I and In → I almost everywhere on D; moreover, the convergence is monotone. Consequently, In → I

in L1 (D),

ω · ∇In = −βIn + sS (In−1 ) + κk 2 F → −βI + sS (I) + κk 2 F

in L1 (D).

Thus, ω · ∇I = −βI + sS (I) + κk 2 F ∈ L∞ (D). Since I ∈ L∞ (D), we have I ∈ W ∞ (D). Since In → I in W 1 (D) and 0  In |Γ−  I, 0  In |Γ+  I, it follows that In |Γ− → I|Γ− in 1 1 (Γ+ ). Therefore, (Γ− ) and In |Γ+ → I|Γ+ in L L − − − − (In |Γ+ ) + Pd,ij (In |Γ+ ) → Rd,ij (I|Γ+ ) + Pd,ij (I|Γ+ ) Rd,ij i

j

i

j

Rd− (In |Γ+ ) + Pd− Bd (In |Γ+ ) → Rd− (I|Γ+ ) + Pd− Bd (I|Γ+ )

1 (Γ− ), in L ij 1 (S − ). in L 173

Passing to the limit as n → ∞ in (5.23), we see that the function I satisfies the equalities (5.2) and (5.3). Thus, I is a solution to the problem (5.1)–(5.3). The existence of a solution I ∈ W ∞ (D) in the case F  0, J∗  0 is proved. If F  0, J∗  0, then the existence of a solution follows from the above argument with F , J∗ , and I replaced by −F , −J∗ , and −I respectively. In the general case, one should use the decomposition F = F+ + F− , J∗ = J∗+ + J∗− , where F+ = max{F, 0}, F− = min{F, 0}, J∗+ = max{J∗ , 0}, J∗− = min{J∗ , 0}, find solutions I+ and I− corresponding to the data F+ , J∗+ and F− , J∗− respectively, and put I = I+ + I− . The uniqueness of a solution follows from Corollary 5.3. ∗

1,p (S − ), p = ∞. Let us prove the existence of a solution in the case F ∈ Lp (D), J∗ ∈ L Theorem 5.4. Let ∂Gj ∈ C 1+λ for all 1  j  m with some λ ∈ (0, 1). Suppose that ∗ 1,p (S − ), where p ∈ (1 + (2λ)−1 , ∞). Then the problem (5.1)–(5.3) has a F ∈ Lp (D) and J∗ ∈ L p (D). unique solution I ∈ W Proof. We set Fn = max{−n, min{F, n}} and J∗,n = max{−n, min{J∗ , n}}, n  1. Since ∗ 1,∞ (S − ), from Theorem 5.3 it follows that for every n  1 the problem Fn ∈ L∞ (D) and J∗,n ∈ L ω · ∇In + βIn = sS (In ) + κk 2 Fn ,

(x, ω) ∈ D,

− − (In |Γ+ ) + Pd,ij (In |Γ+ ), In |Γ− = Rd,ij i

i

(5.24)

(ω, x) ∈ Γ− ij ,

j

i = j

In |Γ− = Rd− (In |Γ+ ) + Pd− Bd (In |Γ+ ) + Pd− Cd (J∗,n ),

(5.25)

(ω, x) ∈ S −

(5.26)

has a unique solution In ∈ W ∞ (D). Since the problem is linear, the estimate (5.9) and (5.10) lead to the estimate

In − I W p (D)  C( Fn − F Lp (D) + J∗,n − J∗, 1,p L

∗

(S − )

)

∀ n  1, ∀  1.

∗

1,p (S − ) as n → ∞, the sequence {In }∞ is a Since Fn → F in Lp (D) and J∗,n → J∗ in L n=1 p Cauchy sequence in W (D). By the completeness of the space W p (D), there exists a function I ∈ W p (D) such that In → I in W p (D) as n → ∞. 1,p (Γ± ). By Lemma 2.3 and Theorem 2.3, the space W p (D) is continuously embedded into L 1,p ± (Γ ). Consequently, Hence In |Γ± → I|Γ± in L − − − − Rd,ij (In |Γ+ ) + Pd,ij (In |Γ+ ) → Rd,ij (I|Γ+ ) + Pd,ij (I|Γ+ ) i

j

i

j

p (Γ− ) and in L ij Rd− (In |Γ+ ) + Pd− Bd (In |Γ+ ) + Pd− Cd (J∗,n ) → Rd− (I|Γ+ ) + Pd− Bd (I|Γ+ ) + Pd− Cd (J∗ ) p (S − ). Passing to the limit in (5.24)–(5.26) as n → ∞, we get the equalities (5.1)–(5.3). in L 1,p (Γ+ ), then I|Γ− ∈ L p (Γ− ) in view of (5.2) and (5.3). Thus, the We emphasize that if I|Γ+ ∈ L p  (D) is proved. existence of a solution I ∈ W The uniqueness of a solution follows from Corollary 5.3. 174

∗

1 (S − ). We note Theorem 5.4 does not cover the case p = 1, i.e., F ∈ L1 (D) and J∗ ∈ L that this case is mathematically difficult, but is important from the physical point of view. We consider this case under two additional assumptions: (A1 ) For any i = j the set ∂Gi ∩ ∂Gj either is empty or has zero measure on ∂G. (A2 ) The quantity α defined in (4.11) satisfies the condition α(x)  α < 1, x ∈ ∂G. Under Assumption (A1 ), the conditions (5.2) are absent and ◦

meas (Γ− \ S − ; dΓ) = meas (S − ; dΓ) = 0. Therefore, the statement of the problem is simplified and takes the form ω · ∇I + βI = sS (I) + κk 2 F,

(ω, x) ∈ D,

(5.27)

I|Γ− = Rd− (I|Γ+ ) + Pd− Bd (I|Γ+ ) + Pd− Cd (J∗ ),

(ω, x) ∈ Γ− .

(5.28)

The simplest form of this problem is realized if G is a convex domain, namely, ω · ∇I + βI = sS (I) + κk 2 F, I|Γ− = Rd− (I|Γ+ ) + Pd− (J∗ ),

(ω, x) ∈ D, (ω, x) ∈ Γ− .

∗

In this case, S − = S − = Γ− ; moreover, Bd (I|Γ+ ) = 0, Cd (J∗ ) = J∗ . We note that α(x) ≡ 0 for a convex domain.. Theorem 5.5. Let Assumptions (A1 ) and (A2 ) be satisfied. If F ∈ Lp (D) and J∗ ∈ ∗ p (D). 1,p (S − ), 1  p < ∞, then the problem (5.27), (5.28) has a unique solution I ∈ W L Proof. We repeat the proof of Theorem 5.4. In view of the inequality (4.1) and the equality + ) = 0, from the estimate (5.9) it follows that meas (Γ+ \ S + ; dΓ 

max kj

1jm

−2p/q

(1 − α)(1 − ρ − ) In |Γ+ − I |Γ+ pp

L (Γ+ )

 Fn − F pLp

(D) κk2

+

1 π p−1

J∗,n − J∗, p

∗

 1,p (S − ) L

p (Γ+ ) and In |Γ+ → I|Γ+ in L p (Γ+ ). for all n  1 and   1. By this inequality, I|Γ+ ∈ L − − − − p − (Γ ). Passing to the Therefore, Rd (In |Γ+ ) + Pd Bd (In |Γ+ ) → Rd (I|Γ+ ) + Pd Bd (I|Γ+ ) in L limit in (5.24) and (5.26) as n → ∞, we obtain (5.27) and (5.28). From (5.28) it follows that p (D) is proved. The solution is unique by p (Γ− ). The existence of a solution I ∈ W I|Γ− ∈ L Corollary 5.3.

Acknowledgment The work is supported by the Russian Foundation for Basic Research (grant No. 1301-00201), the Ministry of Education and Science of the Russian Federation (agreement No. 14.V37.21.0864), and Board grants of the President of the Russian Federation (project NSh2033.2012.1). 175

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Submitted on April 18, 2013

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