GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 5, 1997, 421-438

SOLUTION OF THE BASIC BOUNDARY VALUE PROBLEMS OF STATIONARY THERMOELASTIC OSCILLATIONS FOR DOMAINS BOUNDED BY

SPHERICAL SURFACES

L. GIORGASHVILI

Abstract. The boundary value problems of stationary thermoelastic oscillations are investigated for the entire space with a spherical cavity, when the limit values of a displacement vector and temperature or of a stress vector and heat flow are given on the boundary. Also, consideration is given to the boundary-contact problems when a nonhomogeneous medium fills up the entire space and consists of several homogeneous parts with spherical interface surfaces. Given on an interface surface are differences of the limit values of displacement and stress vectors, also of temperature and heat flow, while given on a free boundary are the limit values of a displacement vector and temperature or of a stress vector and heat flow. Solutions of the considered problems are represented as absolutely and uniformly convergent series.

One of the main methods of solving the spatial problems of elasticity is the Fourier method based on using various representations of solutions of equilibrium equations through harmonic, biharmonic, or metaharmonic functions.

When solving problems by the said method the main difficulty consists in satisfying the boundary conditions. One of the approaches to overcoming this difficulty developed in [1] and [2] is to construct eigenfunctions of vector structure on the boundary.

In this paper, systems of homogeneous equations of stationary thermoelastic oscillations are solved in terms of four metaharmonic functions. Such a representation of solutions enables one to satisfy the boundary conditions quite easily.

1991 Mathematics Subject Classification. 73C02. Key words and phrases. Thermoelasticity, Fourier method.

421

1072-947X/97/0900-0421$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

Some Auxiliary Formulas and Theorems. A system of homogeneous equations of stationary thermoelastic oscillations has the form [3], [4]

μΔu + (λ + μ) grad div u − γ grad u_{4} + ρσ^{2}u = 0,

Δu_{4} + \frac{iσ}{κ} u_{4} + iση div u = 0,

(1)

where Δ is the Laplace operator, u = (u_{1}, u_{2}, u_{3}) the elastic displacement vector, u_{4} the temperature, ρ the medium density, σ the oscillation frequency; λ, μ, η, κ, γ are the constants characterizing the physical properties of the considered elastic body and satisfying the conditions

μ > 0, 3λ + 2μ > 0, γ/η > 0, κ > 0, λ + 2μ ≠ γηκ.

The thermoelastic stress vector is written as [4]

P (∂_{x}, n)U = T (∂x, n)u − γn(x)u_{4},  (2)

where U = (u, u_{4}), T (∂_{x}, n)u is the stress vector of classical elasticity,

T (∂_{x}, n)u = 2μ \frac{∂u}{∂n} + λn div u + μ[n × rot u].

We introduce the notation [5]

\begin{matrix}X_{mk}(θ, φ) = e_{r}Y_{k}^{(m)}(θ, φ), k ≥ 0, \\ Y_{mk}(θ, φ)= \frac{1}{\sqrt{k(k + 1)}}(e_{θ} \frac{∂}{∂θ} + \frac{e_{φ}}{sin θ} \frac{∂}{∂φ})Y_{k}^{(m)}(θ, φ), k ≥1, \\ Z_{mk}(θ, φ)= \frac{1}{\sqrt{k(k + 1)}}(\frac{e_{θ}}{sin θ} \frac{∂}{∂φ} −e_{φ} \frac{∂}{∂θ} )Y_{k}^{(m)}(θ, φ), k ≥1, \end{matrix}  (3)

where |m| ≤ k, e_{r}, and e_{θ}, e_{φ} are the orthogonal unit vectors:

e_{r} = (cos φ sin θ, sin φ sin θ, cos θ), e_{θ} = (cos φ cos θ, sin φ cos θ, − sin θ), e_{φ} = (− sin φ, cos φ, 0),

Y_{k}^{(m)}(θ, φ) = \sqrt{\frac{2k + 1}{4π} · \frac{(k − m)!}{(k + m)!}} P_{k}^{(m)}(cos θ)e^{imφ},

where P_{k}^{(m)}(cos θ) is the adjoint Legendre function of first kind, kth degree, and mth order.

On the sphere r = const, vectors (3) form a complete orthonormal system of vector-functions [5]. Let us show that the following formulas are valid:

\begin{matrix}[e_{r} × X_{mk}(θ, φ)] = 0, [e_{r} × Y_{mk}(θ, φ)] = −Z_{mk}(θ, φ), \\ [e_{r} × Z_{mk}(θ, φ)] = Y_{mk}(θ, φ), \end{matrix}  (4)

grad (Φ(r)Y_{k}^{(m)}(θ, φ)) = \frac{dΦ(r)}{dr} X_{mk}(θ, φ) +

+\frac{\sqrt{k(k + 1)}}{r} Φ(r)Y_{mk}(θ, φ),  (5)
rot[xΦ(r)Y_{k}^{(m)}(θ, φ)] = \sqrt{k(k + 1)} Φ(r)Z_{mk}(θ, φ),  (6)

where Φ(r) is a function of r, x = (x_{1}, x_{2}, x_{3}), (r, θ, φ) are the spherical coordinates of the point x.

If in formulas (3) we set

[e_{r} × e_{r}] = 0, [e_{r} × e_{θ}] = e_{φ}, [e_{r} × e_{φ}] = −e_{θ},

then we will obtain formula (4).

We rewrite the operator grad in terms of spherical coordinates

grad = e_{r} \frac{∂}{∂r} + \frac{1}{r}(e_{θ} \frac{∂}{∂θ} + \frac{e_{φ}}{sin θ} \frac{∂}{∂φ}),

and obtain

grad[Φ(r)Y_{k}^{(m)}(θ, φ)] = (e_{r}Y_{k}^{(m)}(θ, φ)) \frac{dΦ(r)}{dr} +

+\frac{1}{r} Φ(r)(e_{θ} \frac{∂}{∂θ} + \frac{e_{φ}}{sin θ} \frac{∂}{∂φ})Y_{k}^{(m)}(θ, φ) =

= \frac{dΦ(r)}{dr} X_{mk}(θ, φ) + \frac{\sqrt{k(k + 1)}}{r} Φ(r)Y_{mk}(θ, φ),

which proves equality (5). The proof of formula (6) follows from (4), (5) and from the identity

rot [xΦ(r)Y_{k}^{(m)}(θ, φ)] = −[x × grad (Φ(r)Y_{k}^{(m)}(θ, φ))].

In what follows it will be convenient for us to represent the Fourier series of the vector-function f(θ, φ) by system (3) as

f(θ, φ) = α_{00}X_{00}(θ, φ) + ∑_{k=1}^{∞} ∑_{m=−k}^{k} {α_{mk}X_{mk}(θ, φ) +
+ \sqrt{k(k + 1)} [β_{mk}Y_{mk}(θ, φ) + γ_{mk}Z_{mk}(θ, φ)]\},  (7)

424 L. GIORGASHVILI

where α_{mk}, β_{mk}, γ_{mk} are the Fourier coefficients:

α_{mk} = ∫_{0}^{2π}dφ ∫_{0}^{π} f(θ, φ) · \overline{X}_{mk}(θ, φ) sin θ dθ, k ≥ 0,

β_{mk} = \frac{1}{\sqrt{k(k + 1)}} ∫_{0}^{2π}dφ ∫_{0}^{π} f(θ, φ) · \overline{Y}_{mk}(θ, φ) sin θ dθ, k ≥ 1,

γ_{mk} = \frac{1}{\sqrt{k(k + 1)}} ∫_{0}^{2π}dφ ∫_{0}^{π} f(θ, φ) · \overline{Z}_{mk}(θ, φ) sin θ dθ, k ≥ 1,

(8)

\overline{X}_{mk}, \overline{Y}_{mk}, \overline{Z}_{mk} are the vectors complex-conjugated to x_{mk}, y_{mk}, z_{mk}, respectively.

If in formulas (8) we take into account that on the sphere of unit radius

[Y_{mk}(θ, φ)]_{j} = \frac{1}{k(k + 1)} D_{j}Y_{k}^{(m)}(θ, φ),
[Z_{mk}(θ, φ)]_{j} = \frac{1}{k(k + 1)} \frac{∂}{∂s_{j}} Y_{k}^{(m)}(θ, φ),

where

\frac{∂}{∂s_{j}} = (\frac{e_{θ}}{sin θ} \frac{∂}{∂φ} − e_{φ} \frac{∂}{∂θ} )_{j},
D_{j} = (e_{θ} \frac{∂}{∂θ} + \frac{e_{φ}}{sin θ} \frac{∂}{∂φ})_{j}, j = 1, 2, 3,

then we will obtain

α_{mk} = ∫_{0}^{2π}dφ ∫_{0}^{π}(f · e_{r})\overline{Y} _{k}^{(m)}(θ, φ) sin θ dθ, k ≥ 0,

β_{mk} = \frac{1}{\sqrt{k(k + 1)}} ∫_{0}^{2π}dφ ∫_{0}^{π} [2(f · e_{r}) − ∑_{j=1}^{3} D_{j}f_{j}]\overline{Y} _{k}^{(m)}(θ, φ) sin θ dθ,

γ_{mk} = \frac{1}{\sqrt{k(k + 1)}} ∫_{0}^{2π}dφ ∫_{0}^{π} ∑_{j=1}^{3} \frac{∂f_{j}}{∂s_{j}} \overline{Y} _{k}^{(m)}(θ, φ) sin θ dθ.

(9)

Let

F (θ, φ) = ∑_{k=0}^{∞} ∑_{m=−k}^{k} a_{mk}Y_{k}^{(m)}(θ, φ)

be a Fourier series with respect to the orthonormalized system of spherical functions Y_{k}^{(m)}, where

a_{mk} = ∫_{0}^{2π}dφ ∫_{0}^{π} F (θ, φ)\overline{Y} _{k}^{(m)}(θ, φ) sin θ dθ.

The following theorem is true [6].

Theorem 1. If F (y) ∈ C^{(l)}(S), then Fourier coefficients admit the estimates

a_{mk} = O(k^{−l}).

This theorem and formula (9) imply

Theorem 2. If f(y) ∈ C^{(l)}(S), then the Fourier coefficients α_{mk}, β_{mk}, γ_{mk} admit the estimates

α_{mk} = O(k^{−l}), β_{mk} = O(k^{−l−1}), γ_{mk} = O(k^{−l−1}).

Theorem 3. The vectors X_{mk}(θ, φ), Y_{mk}(θ, φ), and Z_{mk}(θ, φ) satisfy the estimates

\begin{matrix}|X_{mk}(θ, φ)| ≤ \sqrt{\frac{2k + 1}{4π}} , \\ |Y_{mk}(θ, φ)| < \sqrt{\frac{2k(k + 1)}{2k + 1}} , \\ |Z_{mk}(θ, φ)| < \sqrt{\frac{2k(k + 1)}{2k + 1}} . \end{matrix}  (10)

Proof. Using the recurrent relations of Legendre polynomials, the vector Y_{mk}(θ, φ) can be represented as

Y_{mk}(θ, φ) = \frac{1}{2\sqrt{k(k + 1)(2k + 1)}} ×

× \{ − e_{1}[\frac{k + 1}{\sqrt{2k − 1}} \sqrt{(k + m)(k + m − 1)} Y_{k−1}^{(m−1)}(θ, φ) +

+ \frac{k}{\sqrt{2k + 3}} \sqrt{(k − m + 1)(k − m + 2)} Y_{k+1}^{(m−1)}(θ, φ)] +

+ e_{2}[\frac{k + 1}{\sqrt{2k − 1}} \sqrt{(k − m)(k − m − 1)} Y_{k−1}^{(m−1)}(θ, φ) +

+ \frac{k}{\sqrt{2k + 3}} \sqrt{(k + m + 1)(k + m + 2)} Y_{k+1}^{(m+1)}(θ, φ)] +

+ 2e_{3}[\frac{k + 1}{\sqrt{2k − 1}} \sqrt{k^{2} − m^{2}} Y_{k−1}^{(m)}(θ, φ) −

426 L. GIORGASHVILI

− \frac{k}{\sqrt{2k + 3}} \sqrt{(k + 1)^{2} − m^{2}} Y_{k+1}^{(m)}(θ, φ)]\},  (11)

where e_{1} = (1, i, 0), e_{2} = (1, −i, 0), e_{3} = (0, 0, 1).

According to [7] we have

|Y_{k}^{(m)}(θ, φ)| ≤ \sqrt{\frac{2k + 1}{4π}}.  (12)

By virtue of the latter inequality formulas (3) and (11) yield

|X_{mk}(θ, φ)| = |e_{r} · Y_{k}^{(m)}(θ, φ)| = |Y_{k}^{(m)}(θ, φ)| ≤ \sqrt{\frac{2k + 1}{4π}},

|Y_{mk}(θ, φ)| < \sqrt{\frac{2k(k + 1)}{2k + 1}},

|Z_{mk}(θ, φ)| = |e_{r} × Y_{mk}(θ, φ)| = |Y_{mk}(θ, φ)| < \sqrt{\frac{2k(k + 1)}{2k + 1}}.

Definition 4. A solution U = (u, u_{4}) of system (1) will be called regular in the domain Ω if U ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}).

The following theorem is true [4].

Theorem 5. A regular solution of equation (1) admits a representation of the form

\begin{aligned}u(x) & = u^{(1)}(x) + u^{(2)}(x) + u^{(3)}(x), \\ u_{4}(x) & = u_{4}^{(1)}(x) + u_{4}^{(2)}(x), \end{aligned}  (13)

where

(Δ + λ_{j}^{2})u^{(j)}(x) = 0, j = 1, 2, 3, (Δ + λ_{j}^{2})u_{4}^{(j)}(x) = 0, j = 1, 2,

rot u^{(j)}(x) = 0, j = 1, 2, div u^{(3)}(x) = 0, λ_{3}^{2} = \frac{ρσ^{2}}{μ} ,

λ_{1}^{2} + λ_{2}^{2} = \frac{ρσ^{2}}{λ + 2μ} + \frac{iσ}{κ} + \frac{iσγη}{λ + 2μ}, λ_{1}^{2} · λ_{2}^{2} = \frac{iσ}{κ} · \frac{ρσ^{2}}{λ + 2μ}.

(14)

For γ ≠ 0 values λ_{1}^{2}, λ_{2}^{2} are complex numbers. Choose values λ_{1} and λ_{2} so that their imaginary parts are positive, i.e., λ_{j} = α_{j} + iβ_{j}, β_{j} > 0, j = 1, 2.

Definition 6. A solution U = (u, u_{4}) of system (1) will be said to satisfy the thermoelastic radiation condition at infinity if

\begin{matrix}u^{(j)}(x) = o(r^{−1}), & \frac{∂u^{(j)}}{∂x_{k}} = O(r^{−2}), \\ u_{4}^{(j)}(x) = o(r^{−1}), & \frac{∂u_{4}^{(j)}}{∂x_{k}} = O(r^{−2}), j = 1, 2, k = 1, 2, 3, \\ u^{(3)}(x) = O(r^{−1}), & \frac{∂u^{(3)}}{∂r} − iλ_{3}u^{(3)} = o(r^{−1}), r = |x|. \end{matrix} (15)

Denote by Ω_{0} a ball bounded by the spherical surface S with center at the origin and radius R. A complement to the set \overline{Ω}_{0} = Ω_{0} ∪ S will be denoted by Ω_{1} = E_{3} \ \overline{Ω}_{0}.

Theorem 7. A regular solution of equation (1) admits, in the domain Ω_{j}, j = 0, 1, a representation of the form

u(x) = grad[Φ_{1}(x) + Φ_{2}(x)] + rot rot(xΦ_{3}) + rot(xΦ_{4}),
u_{4}(x) = c[(k_{1}^{2} − λ_{1}^{2})Φ_{1}(x) + (k_{1}^{2} − λ_{2}^{2})Φ_{2}(x)], (16)

where

\begin{matrix}(Δ + λ_{j}^{2})Φ_{j}(x) = 0, j = 1, 2, (Δ + λ_{3}^{2})Φ_{j}(x) = 0, j = 3, 4, \\ c = (λ + 2μ)/γ, k_{1}^{2} = ρσ^{2}/(λ + 2μ). \end{matrix} (17)

Proof. The vectors u^{(j)}(x), j = 1, 2, 3, satisfying equations (14) admit the following representations [5]:

u^{(j)}(x) = grad Φ_{j}(x), j = 1, 2,
u^{(3)}(x) = rot rot(xΦ_{3}) + rot(xΦ_{4}),

(18)

where Φ_{j}(x), j = 1, 2, 3, 4, are the scalar functions satisfying equations (17).

If the values of the vector u^{(j)}(x), j = 1, 2, 3, from (18) are substituted into (13), we will have

u(x) = grad[Φ_{1}(x) + Φ_{2}(x)] + rot rot(xΦ_{3}) + rot(xΦ_{4}).  (19)

System (1) implies

u_{4}(x) = [\frac{(λ + 2μ)κi}{γσ} (Δ + k_{1}^{2}) − κη] div u.  (20)

The substitution of the values of the vector u(x) from (19) into (20) gives

u_{4}(x) = c[(k_{1}^{2} − λ_{1}^{2})Φ_{1}(x) + (k_{1}^{2} − λ_{2}^{2})Φ_{2}(x)],

which proves that representation (16) is valid. One can immediately prove that the vector (u, u_{4}) represented by formula (16) is a solution of system (1).

Formulation of the Problems. The following problems will be considered: find, in Ω_{1}, a regular vector U = (u, u_{4}) satisfying system (1), the radiation conditions at infinity, and one of the following boundary conditions:

Problem (I)^{−}. {u(z)}^{−} = f(z), {u_{4}(z)}^{−} = f_{4}(z), z ∈ S;

Problem (II)^{−}. \{P (∂_{z}, n)U(z)\}^{−} = f(z), \{\frac{∂u_{4}(z)}{∂n(z)}\}^{−} = f_{4}(z), z ∈ S,

where n(z) is the external normal unit vector with respect to Ω_{0} at the point z ∈ S. Note that n(x) ≡ e_{r}; f(z) = (f_{1}(z), f_{2}(z), f_{3}(z)), f_{j}(z), j = 1, 2, 3, 4, are the given functions.

Problem A. Find in Ω_{j}, j = 0, 1, a regular vector U^{(j)}(x) = (u^{(j)}(x), u_{4}^{(j)}(x)) satisfying the equation

μ_{j}Δu^{(j)} + (λ_{j} + μ_{j}) grad div u^{(j)} − γ_{j} grad u_{4}^{(j)} + ρ_{j}σ_{j}^{2}u^{(j)} = 0,

Δu_{4}^{(j)} + \frac{iσ_{j}}{κ_{j}} u_{4}^{(j)} + iσ_{j}η_{j} div u^{(j)} = 0, j = 0, 1,

(21)

for j = 0, 1, the radiation conditions at infinity for j = 1, and, on the boundary S, the contact conditions

\begin{matrix}\{u^{(0)}(z)\}^{+} − \{u^{(1)}(z)\}^{−} = f^{(0)}(z), \\ \{u_{4}^{(0)}(z)\}^{+} − \{u_{4}^{(1)}(z)\}^{−} = f_{4}^{(0)}(z), \\ \{P^{(0)}(∂_{z}, n)U^{(0)}(z)\}^{+} − \{P^{(1)}(∂_{z}, n)U^{(1)}(z)\}^{−} = f^{(1)}(z), \\ \frac{γ_{0}}{σ_{0}η_{0}} \{\frac{∂u_{4}^{(0)}(z)}{∂n(z)}\}^{+} − \frac{γ_{1}}{σ_{1}η_{1}} \{\frac{∂u_{4}^{(1)}(z)}{∂n(z)}\}^{−} = f_{4}^{(1)}(z), \end{matrix}  (22)

where f^{(j)}(z) = (f_{1}^{(j)}, f_{2}^{(j)}, f_{3}^{(j)}), j = 0, 1, f_{l}^{(j)}(z), l = 1, 2, 3, 4, are the given functions.

Solution of Problems (I)^{−}, (II)^{−}. A solution of these problems is sought for in form (16), where the functions Φ_{j}(x), j = 1, 2, 3, 4, are written as

Φ_{j}(x) = ∑_{k=0}^{∞} ∑_{m=−k}^{k} h_{k}(λ_{j}r)Y_{k}^{(m)}(θ, φ)A_{mk}^{(j)}, j = 1, 2,  (23)

Φ_{j}(x) = ∑_{k=0}^{∞} ∑_{m=−k}^{k} h_{k}(λ_{3}r)Y_{k}^{(m)}(θ, φ)A_{mk}^{(j)}, j = 3, 4,  (24)

where A_{mk}^{(j)}, j = 1, 2, 3, 4, are the unknown constants,

h_{k}(λ_{j}r) = \sqrt{\frac{R}{r}} \frac{H_{k+1/2}^{(1)}(λ_{j}r)}{H_{k+1/2}^{(1)}(λ_{j}R)}, j = 1, 2, 3,  (25)

H_{k+1/2}^{(1)}(x) is Hankel’s function of first kind.

We will impose on the function Φ_{j}(x), j = 3, 4, the condition

∫_{S′} [Φ_{j}(z)]^{−}d_{z}S = 0, j = 3, 4,  (26)

where S′ is the spherical surface with center at the origin and radius R′ (R < R′ < +∞).

If the values of Φ_{j}(x), j = 3, 4, from (24) are substituted into (26), we will have A_{00}^{(j)} = 0, j = 3, 4.

By putting the expression of the vector U = (u, u_{4}) from (16) into (2) we obtain

P (∂_{x}, n)U(x) = 2μ \frac{∂u}{∂r} + μe_{r}[(2λ_{1}^{2} − λ_{3}^{2})Φ_{1}(x) + (2λ_{2}^{2} − λ_{3}^{2})Φ_{2}(x)] −

− ρσ^{2}r(e_{r} \frac{∂}{∂r} − grad )Φ_{3}(x) +

+ μ[e_{r} × grad (r \frac{∂}{∂r} + 1)Φ_{4}(x)].  (27)

If the values of the function Φ_{j}(x), j = 1, 2, 3, 4, from (23), (24) are substituted into (16), (27), we will have by virtue of (5) and (6)

\begin{aligned}u(x) & = u_{00}(r)X_{00}(θ, φ) ∑_{k=1}^{∞} ∑_{m=−k}^{k} \{u_{mk}(r)X_{mk}(θ, φ) + \\ & + \sqrt{k(k + 1)} [v_{mk}(r)Y_{mk}(θ, φ) + w_{mk}(r)Z_{mk}(θ, φ)]\}, \\ u_{4}(x) & = ∑_{k=0}^{∞} ∑_{m=−k}^{k} η_{mk}(r)Y_{k}^{(m)}(θ, φ), \end{aligned} (28)

\begin{aligned}P (∂_{x}, n)U(x) & = a_{00}(r)X_{00}(θ, φ) ∑_{k=1}^{∞} ∑_{m=−k}^{k} \{a_{mk}(r)X_{mk}(θ, φ) + \\ & +\sqrt{k(k + 1)} [b_{mk}(r)Y_{mk}(θ, φ)+c_{mk}(r)Z_{mk}(θ, φ)]\}, \\ \frac{∂u_{4}(x)}{∂n(x)} & = ∑_{k=0}^{∞} ∑_{m=−k}^{k} η′_{mk}(r)Y_{k}^{(m)}(θ, φ), \end{aligned} (29)

where

\begin{aligned}u_{mk}(r) & = ∑_{j=1}^{2} \frac{d}{dr} h_{k}(λ_{j}r)A_{mk}^{(j)} + \frac{k(k + 1)}{r} h_{k}(λ_{3}r)A_{mk}^{(3)}, \\ v_{mk}(r) & = ∑_{j=1}^{2} \frac{1}{r} h_{k}(λ_{j}r)A_{mk}^{(j)} + (\frac{d}{dr} + \frac{1}{r})h_{k}(λ_{3}r)A_{mk}^{(3)}, \\ w_{mk}(r) & = h_{k}(λ_{3}r)A_{mk}^{(4)}, \\ a_{mk}(r) & = 2μ ∑_{j=1}^{2} [\frac{d^{2}}{dr^{2}} + λ_{j}^{2} − \frac{1}{2} λ_{3}^{2}]h_{k}(λ_{j}r)A_{mk}^{(j)} + \\ & + 2μk(k + 1) \frac{d}{dr} [\frac{1}{r} h_{k}(λ_{3}r)]A_{mk}^{(3)}, \\ b_{mk}(r) & = 2μ ∑_{j=1}^{2} \frac{d}{dr} [\frac{1}{r} h_{k}(λ_{j}r)]A_{mk}^{(j)} − \\ & − 2μ[\frac{1}{r} \frac{d}{dr} + \frac{1}{2} λ_{3}^{2} − \frac{k(k + 1) − 1}{r^{2}}]h_{k}(λ_{3}r)A_{mk}^{(3)}, \\ c_{mk}(r) & = μ(\frac{d}{dr} − \frac{1}{r})h_{k}(λ_{j}r)A_{mk}^{(j)}, \\ η_{mk}(r) & = c ∑_{j=1}^{2}(k_{1}^{2} − λ_{j}^{2})h_{k}(λ_{j}r)A_{mk}^{(j)}. \end{aligned}  (30)

Assume that the vector-function f(z) can be expanded into series (7), while the function f_{4}(z) can be expanded with respect to the system of spherical functions Y_{k}^{(m)}(θ, φ):

f_{4}(z) = ∑_{k=0}^{∞} ∑_{m=−k}^{k} δ_{mk}Y_{k}^{(m)}(θ, φ),  (31)

where δ_{mk} are the Fourier coefficients.

Using the boundary conditions of Problems (I)^{−}, (II)^{−} and formulas (28), (29), (7), (31), for the constants A_{mk}^{(j)}, j = 1, 2, 3, 4, we obtain the following systems of algebraic equations:

\begin{matrix}u_{mk}(R) = α_{mk}, η_{mk}(R) = δ_{mk}, k ≥ 0, \\ v_{mk}(R) = β_{mk}, w_{mk}(R) = γ_{mk}, k ≥ 1, \end{matrix}  (32)

for Problem (I)^{−};

\begin{matrix}a_{mk}(R) = α_{mk}, η′_{mk}(R) = δ_{mk}, k ≥ 0, \\ b_{mk}(R) = β_{mk}, c_{mk}(R) = γ_{mk}, k ≥ 1, \end{matrix}  (33)

for Problem (II)^{−}.

Theorem 8. Problems (I)^{−} and (II)^{−} have one solution at most.

Proof. It is enough to show that a regular solution of the homogeneous boundary value Problems (I)_{0}^{−} and (II)_{0}^{−}, satisfying conditions (15), is identically zero. Let U = (u, u_{4}) be a regular solution of Problem (I)_{0}^{−} and (II)_{0}^{−}, satisfying the thermoelastic radiation condition at infinity. We write the Green formula of system (1) in the domain which is bounded by the concentric spheres S and S(0, r), r > R [2]:

\frac{2γ}{iση} ∫_{Ω_{r}} | grad u_{4}|^{2}dx=− ∫_{S} \{\overline{u}·P U −u·P \overline{U} + \frac{γ}{iση} (u_{4} \frac{∂\overline{u}_{4}}{∂n} +\overline{u}_{4} \frac{∂u_{4}}{∂n} )\}^{−}dS+

+ ∫_{S(0,r)} [\overline{u} · P U − u · P \overline{U} + \frac{γ}{iση} (u_{4} \frac{∂\overline{u}_{4}}{∂n} + \overline{u}_{4} \frac{∂u_{4}}{∂n} )]dS.  (34)

Taking into account the boundary conditions of the homogeneous Problems (I)_{0}^{−} and (II)_{0}^{−} in (34), we obtain

\frac{2γ}{iση} ∫_{Ω_{r}} | grad u_{4}|^{2}dx =

= ∫_{S(0,r)} [\overline{u} · P U − u · P \overline{U} + \frac{γ}{iση} (u_{4} \frac{∂\overline{u}_{4}}{∂n} + \overline{u}_{4} \frac{∂u_{4}}{∂n} )]dS.  (35)

Since the imaginary parts of the constants λ_{1} and λ_{2} are positive, Hankel’s function H_{k+1/2}^{(1)}(λ_{j}r) and its complex conjugates \overline{H}_{k+1/2}^{(1)}(λ_{j}r), j = 1, 2, decrease exponentially at infinity. By substituting the values u, P U, u_{4}, \frac{∂u_{4}}{∂n}, from (28), (29) into (35) and using the formulas [7]

\begin{matrix}H_{k+1/2}^{(1)}(λ_{3}r) \frac{d}{dr} H_{k+1/2}^{(2)}(λ_{3}r)−H_{k+1/2}^{(2)}(λ_{3}r) \frac{d}{dr} H_{k+1/2}^{(1)}(λ_{3}r)= \frac{4}{πir} , \\ H_{k+1/2}^{(l)}(λ_{3}r) = O(r^{−1/2}), l = 1, 2, \end{matrix} (36)

we have

\frac{2γ}{iση} lim_{r→∞} ∫_{Ω_{r}} | grad u_{4}|^{2}dx +

+\frac{4μR}{πi} ∑_{k=1}^{∞} ∑_{m=−k}^{k} \frac{k(k + 1)}{|H_{k+1/2}^{(1)}(λ_{3}R)|^{2}} [λ_{3}^{2}|A_{mk}^{(3)}|^{2} + |A_{mk}^{(4)}|^{2}] = 0.

Hence it follows that

grad u_{4}(x) = 0, x ∈ Ω^{−}, A_{mk}^{(j)} = 0, j = 3, 4.  (37)

Taking into account the behavior of u_{4}(x) at infinity and expansion (24), from equality (37) we obtain

u_{4}(x) ≡ 0, Φ_{j}(x) ≡ 0, j = 3, 4, x ∈ Ω_{1}.  (38)

(16) and (38) imply

(k_{1}^{2} − λ_{1}^{2})Φ_{1}(x) + (k_{1}^{2} − λ_{2}^{2})Φ_{2}(x) ≡ 0, x ∈ Ω_{1}.  (39)

Applying the operator Δ + λ_{j}^{2}, j = 1, 2, to both parts of equalities (16) and (38), we have

λ_{j}^{2}(λ_{j}^{2} − k_{1}^{2})Φ_{j}(x) ≡ 0, j = 1, 2, x ∈ Ω_{1}.

Therefore Φ_{j}(x) ≡ 0, j = 1, 2, x ∈ Ω_{1}, and by virtue of equalities (16) and (39) we finally obtain U = (u, u_{4}) ≡ 0.

Remark. A different proof of this theorem is given in [2].

Lemma 9. Formulas (16), (18), (26) establish one-to-one correspondence between the regular solution U = (u, u_{4}) of equations (1) and the system of functions {Φ_{j}(x), j = 1, 2, 3, 4}.

Proof. To prove Lemma 9 it is enough to show that the triviality of the vector U(x) implies the triviality of the functions Φ_{j}(x), j = 1, 2, 3, 4, and vice versa.

Let us express the functions Φ_{j}(x), j = 1, 2, 3, 4, in terms of the components of the vector (u, u_{4}). By formula (16) we have

Φ_{j}(x) = \frac{(−1)^{j}}{k_{1}^{2}(λ_{1}^{2} − λ_{2}^{2})}[(k_{1}^{2} − λ_{3−j}^{2}) div u + \frac{λ_{3−j}^{2}}{c} u_{4}(x)], j = 1, 2,

r(\frac{∂^{2}}{∂r^{2}} + \frac{2}{r} \frac{∂}{∂r} + λ_{2}^{2})Φ_{3}(x) = (e_{r} · u) + \frac{1}{k_{1}^{2}} \frac{∂}{∂r} div u − \frac{1}{ck_{1}^{2}} \frac{∂u_{r}}{∂r} ,

r(\frac{∂^{2}}{∂r^{2}} + \frac{2}{r} \frac{∂}{∂r} + λ_{2}^{2})Φ_{4}(x) = (e_{r} · rot u).

(40)

For u(x) = 0, u_{4}(x) = 0 formulas (40) imply by virtue of condition (26) that Φ_{j}(x), j = 1, 2, 3, 4. Indeed, if u(x) = 0, u_{4}(x) = 0, then it follows from (40) that Φ_{j}(x) = 0, j = 1, 2, and

(\frac{∂^{2}}{∂r^{2}} + \frac{2}{r} \frac{∂}{∂r} + λ_{2}^{2})Φ_{j}(x) = 0, j = 3, 4.  (41)

Applying expansions (24) and equality (26) to this formula, we obtain

∑_{k=1}^{∞} ∑_{m=−k}^{k} \frac{k(k + 1)}{r^{2}} h_{k}(λ_{3}r)Y_{k}^{(m)}(θ, φ)A_{mk}^{(j)} = 0, j = 3, 4,

which gives us A_{mk}^{(j)} = 0, k ≥ 1, and therefore Φ_{j}(x) = 0, j = 3, 4. If Φ_{j}(x) = 0, j = 1, 2, 3, 4, then (16) implies U(x) = 0.

By Theorem 8 and Lemma 9 we conclude that systems (32) and (33) have unique solutions. If the solutions of systems (32) and (41) are put into (28) and (29), respectively, then we shall obtain formal solutions of Problems (I)^{−} and (II)^{−}.

To substantiate the method, first we have to show that series (28) and (29) are convergent. For k → ∞ the following relations are fulfilled [7]:

h_{k}(λ_{j}r) ∼ (\frac{R}{r})^{k+1}, \frac{d}{dr} h_{k}(λ_{j}r) ∼ −\frac{k}{r} (\frac{R}{r})^{k+1}.  (42)

Putting the solution of system (32) (or of (33)) into formulas (30) and taking into account estimates (10) and (42), we find that series (28), (29) are majorized by the series

\begin{matrix}M ∑_{k=k_{0}}^{∞} k^{5/2}(\frac{R}{r})^{k+1}[|α_{mk}| + |δ_{mk}| + k(|β_{mk}| + |γ_{mk}|)], \\ M = const > 0. \end{matrix}  (43)

If x ∈ Ω_{1}, then R < r and series (43) converges. For this series to converge at the boundary, it is enough that the Fourier coefficients α_{mk}, β_{mk}, γ_{mk}, δ_{mk} admit the estimates

α_{mk} = O(k^{−4}), δ_{mk} = O(k^{−4}), β_{mk} = O(k^{−5}), γ_{mk} = O(k^{−5}). (44)

Theorems 1, 2 imply that the Fourier coefficients admit estimates (44) if the vector-function f(z) ∈ C^{4}(S) and the function f_{4}(z) ∈ C^{4}(S).

Substituting the functions Φ_{j}(x), j = 1, 2, 3, 4, from (23), (24) into (18), we obtain

\begin{aligned}u^{(j)}(x) & = \frac{d}{dr} h_{0}(λ_{j}r)X_{00}(θ, φ) ∑_{k=1}^{∞} ∑_{m=−k}^{k} [\frac{d}{dr} h_{k}(λ_{j}r)X_{mk}(θ, φ) + \\ & + \frac{\sqrt{k(k + 1)}}{r} h_{k}(λ_{j}r)Y_{mk}(θ, φ)]A_{mk}^{(j)}, j = 1, 2, \\ u^{(3)}(x) & = ∑_{k=1}^{∞} ∑_{m=−k}^{k} \{[\frac{k(k + 1)}{r} h_{k}(λ_{3}r)X_{mk}(θ, φ) + \\ & + \sqrt{k(k + 1)}(\frac{d}{dr} + \frac{1}{r})h_{k}(λ_{3}r)Y_{mk}(θ, φ)]A_{mk}^{(3)} + \\ & + \sqrt{k(k + 1)} h_{k}(λ_{3}r)Z_{mk}(θ, φ)A_{mk}^{(4)}\}. \end{aligned} (45)

For r → ∞ we have the relations [7]

H_{k+1/2}^{(1)}(λ_{j}r) = \sqrt{\frac{2}{πλ_{j}r}} e^{i(λ_{j}r− \frac{k+1}{2} π)}[1 + O(r^{−1}], j = 1, 2, 3, (46)

|(\frac{d}{dr} − iλ_{3})\frac{H_{k+1/2}^{(1)}(λ_{3}r)}{\sqrt{r}}| ≤ \frac{M_{1}}{r^{2}} , M_{1} = const > 0,  (47)

where M_{1} does not depend on k.

Applying asymptotics (46) and (47) to formulas (45) and (28), by virtue of estimates (44) and the fact that the imaginary parts of λ_{j}, j = 1, 2, are positive, we conclude that the vectors u^{(j)}(x), j = 1, 2, 3, and the function u_{4}(x) satisfy condition (15) at infinity.

Thus the vector U = (u, u_{4}) defined by formula (28), where the unknown constants A_{mk}^{(j)}, j = 1, 2, 3, 4, are a solution of system (32) or (33), is a regular solution of Problem (I)^{−} or (II)^{−}.

Solution of Problem A. A solution of this problem will be sought for in the form

u^{(j)}(x) =grad[Φ_{1}^{(j)}(x) + Φ_{2}^{(j)}(x)] + rot rot(xΦ_{3}^{(j)}(x)) + rot(xΦ_{4}^{(j)}(x)),
u_{4}^{(j)}(x) =c_{j}[(k_{1j}^{2} −λ_{1j}^{2})Φ_{1}^{(j)}(x)+(k_{ij}^{2} −λ_{ij}^{2})Φ_{2}^{(j)}(x)], x∈Ω_{j}, j =0, 1,

(48)

where

c_{j}(λ_{j} + 2μ_{j})/γ_{j}, k_{ij}^{2} = ρ_{j}σ_{j}^{2}/(λ_{j} + 2μ_{j}),
(Δ + λ_{lj}^{2})Φ_{l}^{(j)}(x) = 0, l = 1, 2,
(Δ + λ_{3j}^{2})Φ_{l}^{(j)}(x) = 0, l = 3, 4, j =0, 1.

The constants λ_{lj}, l = 1, 2, 3, j = 0, 1, have form (14), where j corresponds to the domain Ω_{j}.

The functions Φ_{l}^{(j)}(x), l = 1, 2, 3, 4, j = 0, 1, will be sought for in the form

\begin{aligned}Φ_{l}^{(0)}(x) & = ∑_{k=0}^{∞} ∑_{m=−k}^{k} g_{k}(λ_{l0}r)Y_{k}^{(m)}(θ, φ)B_{mk}^{(l)}, \\ Φ_{l}^{(1)}(x) & = ∑_{k=0}^{∞} ∑_{m=−k}^{k} h_{k}(λ_{l1}r)Y_{k}^{(m)}(θ, φ)A_{mk}^{(l)}, l = 1, 2, 3, 4, \end{aligned} (49)

where A_{mk}^{(l)}, B_{mk}^{(l)}, l = 1, 2, 3, 4, are the unknown constants, and λ_{4j} ≡ λ_{3j},

g_{k}(λ_{l0}r) = \sqrt{\frac{R}{r}} \frac{I_{k+1/2}(λ_{l0}r)}{I_{k+1/2}(λ_{l0}R)},  (50)

where I_{k+1/2}(x) is Bessel’s function, and h_{k}(λ_{l1}r) has form (25), where λ_{l} is replaced by λ_{l1}.

The following conditions are imposed on the functions Φ_{l}^{(j)}(x), l = 3, 4, j = 0, 1:

∫_{S′} [Φ_{l}^{(0)}(z)]^{+}dS = 0, l = 3, 4,  (51)

∫_{S′′} [Φ_{l}^{(1)}(z)]^{−}dS = 0, l = 3, 4,  (52)

where S′ and S′′ are the spheres with center at the origin and radii R′ and R′′ (0 < R′ < R < R′′ < +∞), respectively.

If the functions Φ_{l}^{(j)}(x) from (49) are inserted into (51) and (52), we will obtain A_{00}^{(l)} = 0, B_{00}^{(j)}, l = 3, 4.

By substituting the function Φ_{l}^{(j)}(x), l = 1, 2, 3, 4, j = 0, 1, from formula (49) into (48) and (27) we obtain

u^{(j)}(x) = u_{00}^{(j)}(r)X_{00}(θ, φ) ∑_{k=1}^{∞} ∑_{m=−k}^{k} {u_{mk}^{(j)}(r)X_{mk}(θ, φ) +
+ \sqrt{k(k + 1)} [v_{mk}^{(j)}(r)Y_{mk}(θ, φ) + w_{mk}^{(j)}(r)Z_{mk}(θ, φ)]\}, (53)

u_{4}^{(j)}(x) = ∑_{k=0}^{∞} ∑_{m=−k}^{k} η_{mk}^{(j)}(r)Y_{k}^{(m)}(θ, φ),

P^{(j)}(∂_{x}, n)U^{(j)}(x) = a_{00}^{(j)}(r)X_{00}(θ, φ) ∑_{k=1}^{∞} ∑_{m=−k}^{k} {a_{mk}^{(j)}(r)X_{mk}(θ, φ) +
+\sqrt{k(k + 1)} [b_{mk}^{(j)}(r)Y_{mk}(θ, φ)+c_{mk}^{(j)}(r)Z_{mk}(θ, φ)]\}, j =0, 1,  (54)

where the expressions for u_{mk}^{(1)}, v_{mk}^{(1)}, …, η_{mk}^{(1)} are given by formulas (30) if the constants λ, μ, …, σ there are replaced by λ_{1}, μ_{1}, …, σ_{1}, while the expressions for u_{mk}^{(0)}, …, η_{mk}^{(0)} are obtained from (30) if the constants λ, μ, …, σ there are replaced by λ_{0}, μ_{0}, …, σ_{0} and Hankel’s function by Bessel’s function.

Let the functions f_{4}^{(j)}(z) and the vector-functions f^{(j)}(z), j = 0, 1, be expanded into the series

\begin{aligned}f^{∗j)}(z) & = α_{00}^{(j)}X_{00}(θ, φ) ∑_{k=1}^{∞} ∑_{m=−k}^{k} \{α_{mk}^{(j)}X_{mk}(θ, φ) + \\ & + \sqrt{k(k + 1)} [β_{mk}^{(j)}Y_{mk}(θ, φ) + γ_{mk}^{(j)}Z_{mk}(θ, φ)]\}, \\ f_{4}^{∗j)}(z) & = ∑_{k=0}^{∞} ∑_{m=−k}^{k} δ_{mk}^{(j)}Y_{k}^{(m)}(θ, φ), j = 0, 1. \end{aligned}  (55)

Using the contact conditions (22) and formulas (53)–(55), we obtain the following system of algebraic equations:

\begin{matrix}u_{mk}^{(0)}(R) − u_{mk}^{(1)}(R) = α_{mk}^{(0)}, η_{mk}^{(0)}(R) − η_{mk}^{(1)}(R) = δ_{mk}^{(0)}, k ≥ 0, \\ v_{mk}^{(0)}(R) − v_{mk}^{(1)}(R) = β_{mk}^{(0)}, w_{mk}^{(0)}(R) − w_{mk}^{(1)}(R) = γ_{mk}^{(0)}, k ≥ 1, \\ a_{mk}^{(0)}(R) − a_{mk}^{(1)}(R) = α_{mk}^{(1)}, \frac{γ}{σ_{0}η_{0}} \frac{d}{dR} η_{mk}^{(0)}(R) − \\ − \frac{γ_{1}}{σ_{1}η_{1}} \frac{d}{dR} η_{mk}^{(1)}(R) = δ_{mk}^{(1)}, k ≥ 0, \\ b_{mk}^{(0)}(R) − b_{mk}^{(1)}(R) = β_{mk}^{(1)}, c_{mk}^{(0)}(R) − c_{mk}^{(1)}(R) = γ_{mk}^{(0)}, k ≥ 1. \end{matrix} (56)

Theorem 10. The homogeneous problem (A)_{0} has only a trivial solution.

Proof. We write Green’s formulas for system (21) in the domains Ω_{0} and Ω_{r}, where the latter is bounded by the concentric surfaces S and S(0, r), r > R [4]:

\frac{2γ_{0}}{iσ_{0}η_{0}} ∫_{Ω_{0}} | grad u_{4}^{(0)}(x)|^{2}dx = ∫_{S} \{\overline{u}^{(0)} · P^{(0)}U^{(0)} − u^{(0)} · P^{(0)}\overline{U}^{(0)} +

+\frac{γ_{0}}{iσ_{0}η_{0}} (u_{4}^{(0)} \frac{∂\overline{u}_{4}^{(0)}}{∂n} + \overline{u}_{4}^{(0)} \frac{∂u_{4}^{(0)}}{∂n} )\}^{+}dS,  (57)

\frac{2γ_{1}}{iσ_{1}η_{1}} ∫_{Ω_{r}} | grad u_{4}^{(1)}(x)|^{2}dx = − ∫_{S} \{\overline{u}^{(1)} · P^{(1)}U^{(1)} − u^{(1)} · P^{(1)}\overline{U}^{(1)} +

+\frac{γ_{1}}{iσ_{1}η_{1}} (u_{4}^{(1)} \frac{∂\overline{u}_{4}^{(1)}}{∂n} + \overline{u}_{4}^{(1)} \frac{∂u_{4}^{(1)}}{∂n} )\}^{+}dS + ∫_{S(0,r)} \{\overline{u}^{(1)} · P^{(1)}U^{(1)} −

−u^{(1)} · P^{(1)}\overline{U}^{(1)} + \frac{γ_{1}}{iσ_{1}η_{1}} (u_{4}^{(1)} \frac{∂\overline{u}_{4}^{(1)}}{∂n} + \overline{u}_{4}^{(1)} \frac{∂u_{4}^{(1)}}{∂n} )\}^{+}dS.  (58)

Applying the homogeneous boundary condition of Problem (A)_{0} to (57) and (59), we obtain

\frac{2γ_{0}}{iσ_{0}η_{0}} ∫_{Ω_{0}} | grad u_{4}^{(0)}(x)|^{2}dx + \frac{2γ_{1}}{iσ_{1}η_{1}} ∫_{Ω_{r}} | grad u_{4}^{(0)}(x)|^{2}dx =

= ∫_{S(0,r)} \{\overline{u}^{(1)} · P^{(1)}U^{(1)} − u^{(1)} · P^{(1)}\overline{U}^{(1)} +

+\frac{γ_{1}}{iσ_{1}η_{1}} (u_{4}^{(1)} \frac{∂\overline{u}_{4}^{(1)}}{∂n} + \overline{u}_{4}^{(1)} \frac{∂u_{4}^{(1)}}{∂n} )\}dS.  (59)

Substituting the expressions for u^{(1)}, P^{(1)}U^{(1)}, u_{4}^{(1)}, and \frac{∂u_{4}^{(1)}}{∂n} from (53),

(54) into (59), and taking into account formula (36) and the fact that the vectors X_{mk}, Y_{mk}, Z_{mk} are normalized, we have

\frac{2γ_{0}}{iσ_{0}η_{0}} ∫_{Ω_{0}} | grad u_{4}^{(0)}(x)|^{2}dx + \frac{2γ_{1}}{iσ_{1}η_{1}} ∫_{Ω_{r}} | grad u_{4}^{(0)}(x)|^{2}dx +

+\frac{4μR}{πi} ∑_{k=1}^{∞} ∑_{m=−k}^{k} \frac{k(k + 1)}{|H_{k+1/2}(λ_{3}R)|^{2}} [λ_{31}^{2}|A_{mk}^{(3)}|^{2} + |A_{mk}^{(4)}|^{2}] = o(1). (60)

Hence, passing to the limit as r → ∞, we find

u_{4}^{(j)}(x) = const, j = 0, 1, A_{mk}^{(l)} = 0, l = 3, 4, k ≥ 1.

Since u_{4}^{(j)}(x) is a metaharmonic function, we have u_{4}^{(j)} ≡ 0, x ∈ Ω_{j}, j = 0, 1. By the equality A_{mk}^{(l)} = 0, l = 3, 4, it follows that Φ_{l}^{(1)}(x) ≡ 0, x ∈ Ω_{1}, l = 3, 4. Using the equality u_{4}^{(1)}(x) ≡ 0, x ∈ Ω_{1}, in representation (48), we obtain Φ_{l}^{(1)} ≡ 0, x ∈ Ω_{1}, l = 1, 2. Thus we have shown that Φ_{l}^{(1)}(x) ≡ 0, x ∈ Ω_{1}, l = 1, 2, 3, 4. By virtue of these equalities we conclude that

u^{(1)}(x) ≡ 0, u_{4}^{(1)} ≡ 0, x ∈ Ω_{1}.  (61)

Using the contact conditions of Problem (A)_{0} and equalities (63), we find

\begin{matrix}\{u^{(0)}(z)\}^{+} = 0, \{u_{4}^{(0)}(z)\}^{+} = 0, \{P^{(0)}U^{(0)}(z)\}^{+} = 0, \\ \{\frac{∂u_{4}^{(0)}}{∂n} \}^{+} = 0, z ∈ S. \end{matrix}  (62)

A general representation of regular solutions of the homogeneous equation (21) in the domain Ω_{0} has the form [4]

2U^{(0)}(x) = ∫_{S} {Γ^{(0)}(x − y, σ_{0})[R^{(0)}U^{(0)}]^{+} −

− [R̃Γ̃^{(0)}(y − x, σ_{0})]′[U^{(0)}(y)]^{+}}d_{y}S,  (63)

where Γ^{(0)}(x − y, σ_{0}) is the fundamental solution of system (21), Γ̃^{(0)}(x − y, σ_{0}) is the matrix of fundamental solutions of the adjoint homogeneous system, RU = (P U, \frac{∂u_{4}}{∂n}), R̃U = (T u − iσηu_{4}, \frac{∂u_{4}}{∂n}).

From (62) and (63) we finally obtain

u^{(0)}(x) ≡ 0, u_{4}^{(0)}(x) ≡ 0, x ∈ Ω_{0}.

By the uniqueness theorem and Lemma 9 we conclude that system (56) has a unique solution. After putting this solution into (53) , we obtain a formal solution of Problem A.

If f^{(j)}(z) ∈ C^{4}(S), f_{4}^{(j)}(z) ∈ C^{4}(S), j = 0, 1, then the constructed formal series (53) gives a regular solution of the problem posed.

References

1. A. F. Ulitko, The method of vector egenfunctions in spatial problems of the elasticity theory. (Russian) Naukova Dumka, Kiev, 1979.

2. L. G. Giorgashvili, Solution of the basic boundary value problems of the theory of elasticity for a ball. (Russian) Proc. I. Vekua Inst. Appl. Math. (Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy) 10(1981), 32–36.

3. W. Novacki, Dynamical problems of thermoelasticity. (Polish) PWN, Warszawa, 1996; Russian Translation: Mir, Moscow, 1970.

4. V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. (Translated from Russian) North-Holland series in applied mathematics and mechanics, v. 25, North-Holland Publishing Company, Amsterdam–New York–Oxford, 1979; Russian original: Nauka, Moscow, 1976.

5. F. M. Mors and G. Feshbakh, Methods of theoretical physics, v. 2. (Russian), Moscow, 1958.

6 S. G. Mikhlin, Multi-dimensional singular integrals and integral equations. (Russian) Fizmatgiz, Moscow, 1962.

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(Received 17.03.1994; revised 10.10.1996)

Author’s address:

Department of Mathematics (99)

Georgian Technical University

77, M. Kostava St., Tbilisi 380075

Georgia