GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 1, 1997, 1-18

NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS OF GENERALIZED ELASTOTHERMODIFFUSION

T. BURCHULADZE

Abstract. The initial boundary value problems with mixed boundary conditions are considered for a system of partial differential equations of generalized electrothermodiffusion. Approximate solutions are constructed and a mathematical substantiation of the method is given.

It is well known that a wide range of problems of mathematical physics includes boundary value and initial boundary value problems for differential equations. Such problems are rather difficult to solve because of an enormous variety of geometrical forms of the investigated objects and the complexity of boundary conditions. Hence an important and timely task has arisen to develop sufficiently effective methods and tools for solving the above-mentioned problems and obtaining their numerical solutions.

Until recently no methods were known for solving problems of elasticity by means of conjugate fields. However, in the past few years there have appeared and keep on appearing numerous published works dedicated to this topic. The results of our studies in this direction are presented mainly in [1, 2, 3].

In this paper, using the generalized Green–Lindsay theory of elastothermodiffusion as an example, the approaches to the solution of nonstationary initial boundary value contact problems with mixed boundary conditions are described for a system of partial differential equations of this theory in a nonhomogeneous medium. These approaches are based on the method of discrete singularities known as the Riesz–Fischer–Kupradze method. The particular case is treated in [4].

1

1991 Mathematics Subject Classification. 73B30, 73C25.

Key words and phrases. Dynamic problems of elastothermodiffusion, Laplace transform, a singular fundamental solution, Riesz–Fischer–Kupradze method of approximate solution.

1072-947X/97/0100-0001$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

Let us consider a three-dimensional isotropic elastic medium in which the thermodiffusive process is taking place. The deformed state is described by the displacement vector v(x, t) = v = (v_{1}, v_{2}, v_{3})^{T} = ∥v_{k}∥_{3×1}, temperature change v_{4}(x, t), and “chemical potential of the medium” v_{5}(x, t). Here x = (x_{1}, x_{2}, x_{3}) is a point of the Euclidean space ℝ^{3}, t the time, C(x, t) = γ_{2} div v(x, t) + a_{12}v_{4}(x, t) + a_{2}v_{5}(x, t), C(x, t) the diffusing substance concentration, and T denotes the operation of transposition.

The object of our investigation is a system of partial differential equations of the generalized theory of elastothermodiffusion having the form [4]

L(\frac{∂}{∂x}, \frac{∂}{∂t})V (x, t) = H(x, t),  (1)

where

V = (v_{1}, v_{2}, …, v_{5})^{T} = (v, v_{4}, v_{5})^{T} = ∥v_{k}∥_{5×1}

is an unknown vector V

LV = ∥(LV)_{k}∥_{5×1},

(LV)_{k} = \begin{cases}μΔv_{k} + (λ + μ)\frac{∂}{∂x_{k}} div v − ρ\frac{∂^{2}v}{∂t^{2}} + \\ − ∑_{l=1}^{2} γ_{l}(1 + τ^{1} \frac{∂}{∂t})\frac{∂v_{3+l}}{∂x_{k}} , k = 1, 2, 3, \\ δ_{k−3}Δv_{k} − a_{k−3}(1 + τ^{0} \frac{∂}{∂t})\frac{∂v_{k}}{∂t} − γ_{k−3} \frac{∂}{∂t} div v− \\ −a_{12}(1 + τ^{0} \frac{∂}{∂t})\frac{∂v_{9−k}}{∂t} , k = 4, 5; \end{cases}

H(x, t) = (X(x, t), X_{4}(x, t), X_{5}(x, t))^{T} = (X_{1}, …, X_{5})^{T} = ∥X_{k}∥_{5×1} is a given vector-function; the elastic, thermal, diffusive, and relaxational constants τ^{0}, τ^{1} satisfy the natural restrictions [3]:

μ > 0, 3λ + 2μ > 0, ρ > 0, a_{k} > 0, δ_{k} > 0, γ_{k} > 0, k = 1, 2,

a_{1}a_{2} − a_{12}^{2} > 0, τ^{1} ≥ τ^{0} > 0.

In particular, the classical case is realized for τ^{1} = τ^{0} = 0.

In this theory the differential operator of stresses which is a 5 × 5 matrix is determined by the formula

R_{τ} (\frac{∂}{∂x}, \frac{∂}{∂t}, n(x))V (x, t) =

= (T (\frac{∂}{∂x}, n(x))v(x, t) − n(x) ∑_{l=1}^{2} γ_{l}(1 + τ^{1} \frac{∂}{∂t})v_{3+l}, δ_{1} \frac{∂v_{4}}{∂n} , δ_{2} \frac{∂v_{5}}{∂n} )^{T} ,

NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS 3

where T (\frac{∂}{∂x}, n(x)) ≡ ∥μδ_{jk} \frac{∂}{∂n} + λn_{j} \frac{∂}{∂x_{k}} + μn_{k} \frac{∂}{∂x_{j}} ∥_{3×3} is the matrix differential operator of elastic stresses in the classical theory [1], n(x) = (n_{1}(x), n_{2}(x), n_{3}(x)) the unit vector δ_{jk} the Kronecker symbol.

Clearly,

R_{τ} (\frac{∂}{∂x}, \frac{∂}{∂t}, n(x))V ≡ R_{τ} (\frac{∂}{∂x}, 0, n(x))V − ∑_{k=1}^{2} r^{k}(τ)\frac{∂v_{3+k}}{∂t} ,

where r^{k}(τ) = γ_{k}τ^{1}(n; 0, 0)^{T}.

Assume that D is a finite domain with the boundary S which is a compact closed surface of the Liapunov class Λ_{2}(α), α > 0 [1]; D_{11} ⊂ D and D_{12} ⊂ D are the nonintersecting domains with the Liapunov boundaries S_{11} and S_{12}, respectively; D_{21} and D_{22} are the nonintersecting finite domains outside S with the boundary Liapunov surfaces S_{21} and S_{22}, respectively; D_{1} is a finite domain bounded by the surfaces S ∪ S_{11} ∪ S_{12}; D_{2} is an infinite domain bounded by the surfaces S ∪S_{21} ∪S_{22}. Consider the infinite domain

D_{1} ∪ S ∪ D_{2} ≡ ℝ^{3}\ ∪^{2}_{l,k=1} \overline{D}_{lk}; S is the contacting surface.

Let D_{1} and D_{2} be two different homogeneous isotropic physical media. The elastothermodiffusive constants for D_{j} will be denoted by the lower left indices _{j}λ,_{j}μ,_{j}ρ,_{j}γ_{l}, …,_{j}τ^{1},_{j}τ^{0}, … while the differential operators by jL(\frac{∂}{∂x}, \frac{∂}{∂t}),_{j}R_{τ} (\frac{∂}{∂x}, \frac{∂}{∂t}, n(x)), …, where j = 1, 2.

Problem A^{t}. In the infinite cylinder

Z_{∞}^{1,2} = {(x, t) : x ∈ D_{1} ∪ D_{2}, t ∈]0, ∞[}

determine a regular vector V = (v, v_{4}, v_{5})^{T} : Z_{∞}^{1,2} → ℝ^{5} of the class C^{1}(\overline{Z}_{∞}^{1,2}) ∩ C^{2}(Z_{∞}^{1,2}) using the boundary conditions

∀(x, t) ∈ Z_{∞}^{1,2} : _{j}L(\frac{∂}{∂x}, \frac{∂}{∂t})V (x, t) = _{j}H(x, t), x ∈ D_{j}, j = 1, 2,
∀x ∈ D_{j} : lim_{t→+0} V (x, t) = _{j}φ^{(0)}(x),

lim_{t→+0} \frac{∂V (x, t)}{∂t} = _{j}φ^{(1)}(x), j = 1, 2,

∀y ∈ S, t ≥ 0 : [V]±_{S} ≡ V^{+}(y, t) − V^{−}(y, t) = f(y, t)
[R_{τ} V ]_{S}^{±} ≡ [_{1}R_{τ} (\frac{∂}{∂y} , \frac{∂}{∂t}, n)V ]^{+} −  (2)
−[_{2}R_{τ} (\frac{∂}{∂y} , \frac{∂}{∂t}, n)V (y, t)]^{−} = F (y, t),
∀y ∈ S_{j1}, t ≥ 0 : V^{+}(y, t) = _{j}f(y, t), j = 1, 2,

∀y ∈ S_{j2}, t ≥ 0 : [_{j}R_{τ} (\frac{∂}{∂y} , \frac{∂}{∂t}, n)V ]^{+} = _{j}F (y, t), j = 1, 2;

for large t and x ∈ D_{2}:

|D_{x,t}^{α}V (x, t)| ≤ \frac{const e^{σ_{0}t}}{1 + |x|^{1+|α|}} , |α| = \overline{0, 2}, σ_{0} ≥ 0,

D_{x,t}^{α} ≡ \frac{∂^{|α|}}{∂x_{1}^{α_{1}}∂x_{2}^{α_{2}}∂x_{3}^{α_{3}}∂t^{α_{4}}} , |α| = ∑_{k=1}^{4} α_{4},

where α = (α_{1}, α_{2}, α_{3}, α_{4}) is the multi-index.

Here

V^{+}(y, t) = lim_{D_{1}∋x→y∈S} V (x, t),
V^{−}(y, t) = lim_{D_{2}∋x→y∈S} V (x, t),
[_{1}R_{τ} V ]^{+} = lim_{D_{1}∋x→y∈S} _{1}R_{τ} (\frac{∂}{∂x}, \frac{∂}{∂t}, n(y))V (x, t),
[_{2}R_{τ} V ]^{−} = lim_{D_{2}∋x→y∈S} _{2}R_{τ} (\frac{∂}{∂x}, \frac{∂}{∂t}, n(y))V (x, t).

In a similar manner we determine the limit values on the surfaces S_{jk}, j, k = 1, 2; here _{j}φ^{(0)},_{j}φ^{(1)} : D_{j} → ℝ^{5}, f, F : S × [0, ∞[→ ℝ^{5}, _{j}H : Z_{∞}^{1,2} → ℝ^{5}, _{j}f : S_{j1} × [0, ∞[→ ℝ^{5}, _{j}F : S_{j2} × [0, ∞[→ ℝ^{5} are given real-valued vector-functions of the definite classes [6].

For a classical (regular) solution to exist it is necessary that the condition of “natural agreement” of the initial data be fulfilled. Here these conditions have the form

∀y ∈ S : _{1}φ^{(0)}(y) −_{2}φ^{(0)}(y) = f(y, 0),

1φ^{(1)}(y) −_{2}φ^{(1)}(y) = \frac{∂f(y, t)}{∂t}|_{t=0},

[_{1}R_{τ} (\frac{∂}{∂y} , 0, n)_{1}φ^{(0)}(y) − ∑_{k=1}^{2}_{1}r^{k}(τ)_{1}φ_{3+k}^{(1)}(y)] −

−[_{2}R_{τ} (\frac{∂}{∂y} , 0, n)_{2}φ^{(0)}(y) − ∑_{k=1}^{2}_{2}r^{k}(τ)_{2}φ_{3+k}^{(1)}(y)] = F (y, 0), (3)

∀y ∈ S_{j1} : _{j}φ^{(0)}(y) = _{j}f(y, 0), _{j}φ^{(1)}(y) = \frac{∂_{j}f(y, t)}{∂t}|_{t=0}, j = 1, 2,
∀y ∈ S_{j2} : _{j}R_{τ} (\frac{∂}{∂y} , 0, n)_{j}φ^{(0)}(y) −

− ∑_{k=1}^{2}_{j}r^{k}(τ)_{j}φ_{3+k}^{(1)}(y) = _{j}F (y, 0), j = 1, 2.

To investigate the dynamic problem A^{t} we shall use the Laplace transform with respect to t. However, in our case these “natural conditions” are not sufficient for substantiating the method. So we must additionally use the “higher order” agreement conditions [2, 3] (with m = \overline{1, 7}):

\begin{aligned}∀y ∈ S : \frac{∂^{m}f(y, t)}{∂t^{m}} |_{t=0} & = _{1}φ^{(m)}(y) −_{2}φ^{(m)}(y), \\ \frac{∂^{m}F (y, t)}{∂t^{m}} |_{t=0} & = [_{1}R_{τ} (\frac{∂}{∂y} , 0, n)_{1}φ^{(m)}(y) − \\ & − ∑_{k=1}^{2}_{1}r^{k}(τ)_{1}φ_{3+k}^{(m+1)}(y)] − \\ & − [_{2}R_{τ} (\frac{∂}{∂y} , 0, n)_{2}φ^{(m)}(y) − \\ & − ∑_{k=1}^{2}_{2}r^{k}(τ)_{2}φ_{3+k}^{(m+1)}(y)], \\ ∀y ∈ S_{j1} : \frac{∂^{m}_{j}f(y, t)}{∂t^{m}} |_{t=0} & = _{j}φ^{(m)}(y), j = 1, 2, \\ ∀y ∈ S_{j2} : \frac{∂^{m}_{j}F (y, t)}{∂t^{m}} |_{t=0} & = _{j}R_{τ} (\frac{∂}{∂y} , 0, n)_{j}φ^{(m)}(y) − \\ & − ∑_{k=1}^{2}_{j}r^{k}(τ)_{j}φ_{3+k}^{(m+1)}(y), j = 1, 2, \end{aligned} (4)

where (m ≥ 2)

(_{j}φ_{1}^{(m)}(x),_{j}φ_{2}^{(m)}(x),_{j}φ_{3}^{(m)}(x))^{T} =
= _{j}ρ^{−1}[_{j}μΔ(_{j}φ_{1}^{(m−2)},_{j}φ_{2}^{(m−2)},_{j}φ_{3}^{(m−2)})^{T} +
+(_{j}λ +_{j}μ) grad div (_{j}φ_{1}^{(m−2)},_{j}φ_{2}^{(m−2)},_{j}φ_{3}^{(m−2)})^{T} −
− ∑_{k=1}^{2}_{j}γ_{k} grad_{j}φ_{3+k}^{(m−2)} − ∑_{k=1}^{2}_{j}γ_{k}_{j}τ^{1} grad_{j}φ_{3+k} − \frac{∂^{m−2}_{j}X}{∂t^{m−2}} |_{t=0}],
ja_{1}_{j}τ^{0}_{j}φ_{4}^{(m)}(x) +_{j}a_{12}_{j}τ^{0}_{j}φ_{5}^{(m)}(x) =  (5)
= _{j}δ_{1}Δ_{j}φ_{4}^{(m−2)}(x) −_{j}a_{1}_{j}φ_{4}^{(m−1)}(x) −_{j}a_{12}_{j}φ_{5}^{(m−1)}(x) −

−_{j}γ_{1} div (_{j}φ_{1}^{(m−1)},_{j}φ_{2}^{(m−1)},_{j}φ_{3}^{(m−1)})^{T}− \frac{∂^{m−2}_{j}X_{4}(x, t)}{∂t^{m−2}} |_{t=0},
ja_{12}_{j}τ^{0}_{j}φ_{4}^{(m)}(x) +_{j}a_{2}_{j}τ^{0}_{j}φ_{5}^{(m)}(x) =
= _{j}δ_{2}Δ_{j}φ_{5}^{(m−2)}(x) −_{j}a_{2}_{j}φ_{5}^{(m−1)}(x) −_{j}a_{12}_{j}φ_{4}^{(m−1)}(x) −
−_{j}γ_{2} div (_{j}φ_{1}^{(m−1)},_{j}φ_{2}^{(m−1)},_{j}φ_{3}^{(m−1)})^{T}− \frac{∂^{m−2}_{j}X_{5}(x, t)}{∂t^{m−2}} |_{t=0}.

Using the method of potential and the theory of multidimensional singular integral equations developed in [1, 2, 3], it can be proved with the aid of the integral Laplace transform, that the structural conditions of agreement (3), (4) are sufficient for the existence of a classical solution of the dynamic problem A^{t}. The proof is similar to that for the case of a homogeneous medium [3, 5]. Here our aim is to construct a solution agorithm for the above-indicated composite nonhomogeneous medium bounded by several closed surfaces and, using this example, we shall realize the method of an approximate construction of Riesz–Fischer–Kupradze solutions (the method of discrete singularities). Incidentally, in the limiting case this method can serve as a tool for proving existence theorems.

Theorem 1. If the initial data of Problem A^{t} satisfy the agreement conditions (3), (4) and certain smoothness conditions (see [6]), then there exists a unique classical solution of the dynamic problem A^{t}, and in the complex half-plane Re ζ > σ_{0}^{∗} this solution is represented by the Laplace–Mellin integral

V (x, t) = \frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞}e^{ζt}V̂(x, ζ)dζ,

where V̂(x, ζ) is a solution of the corresponding elliptic problem, which is represented as a series

V̂(x, ζ) = ∑_{k=1}^{∞} a_{k}(ζ)_{k}U(x, ζ) + U(x, ζ).

This series converges uniformly (with respect to the metric of the space C) in the domain x ∈ \overline{D}′ ⋐ D_{1} ∪ D_{2}; a_{k}(ζ), _{k}U(x, ζ), U(x, ζ) are the known functions and vector-functions (constructed explicitly), ζ = σ + iq, σ > σ_{0}^{∗} > σ_{0} and σ_{0}^{∗} is the known constant.

By the formal application of the Laplace transform

V̂(x, ζ) = ∫_{0}^{∞}e^{−ζt}V (x, t)dt  (6)

NONSTATIONARY INITIAL BOUNDARY VALUE CONTACT PROBLEMS 7

the dynamic problem A^{t} is reduced to the corresponding elliptic problem Â(ζ) with a complex parameter ζ (the spectral problem) for the image V̂(x, ζ):

Problem Â(ζ). Determine a regular vector

V̂(x, ζ) = (v̂, v̂_{4}, v̂_{5})^{T} , ∀ζ ∈ Π_{σ_{0}^{∗}} ≡ {ζ : Re ζ > σ_{0}^{∗} > σ_{0}},

V̂ ∈ C^{1}(\overline{D}_{1} ∪ \overline{D}_{2}) ∩ C^{2}(D_{1} ∪ D_{2}),

in D_{1} ∪ D_{2} using the conditions

∀x ∈ D_{j} : _{j}L(\frac{∂}{∂x}, ζ)V̂(x, ζ) = _{j}H̃(x, ζ), j = 1, 2,  (7)
\begin{matrix}∀y ∈ S : & [V̂]_{S}^{±} ≡ V̂^{+}(y, ζ) − V̂^{−}(y, ζ) = f̂(y, ζ), \\ [R_{τ} V̂]_{S}^{±} ≡ [_{1}R_{τ} (\frac{∂}{∂y} , ζ, n)V̂(y, ζ)]^{+} − \\ − & [_{2}R_{τ} (\frac{∂}{∂y} , ζ, n)V̂(y, ζ)]^{−} = F̃(y, ζ), \end{matrix}  (8)
∀y ∈ S_{j1} : V̂^{+}(y, ζ) = _{j}f̂(y, ζ), j = 1, 2,  (9)
∀y ∈ S_{j2} : [_{j}R_{τ} (\frac{∂}{∂y} , ζ, n)V̂^{+}(y, ζ)]^{+} = _{j}F̃(y, ζ), j = 1, 2,  (10)

where

H̃ = (_{j}X̃,_{j}X̃_{4},_{j}X̃_{5})^{T} ,
jX̃ = _{j}X̂ −_{j}ρ(_{j}φ_{1}^{(1)},_{j}φ_{2}^{(1)},_{j}φ_{3}^{(1)})^{T} −_{j}ρζ(_{j}φ_{1}^{(0)},_{j}φ_{2}^{(0)},_{j}φ_{3}^{(0)})^{T} −

−_{j}τ^{1} ∑_{k=1}^{2} γ_{k} grad_{j}φ_{3+k}^{(0)},

jX̃_{4} = _{j}X̂_{4} −_{j}a_{1}jφ_{4}^{(0)}(x) −_{j}a_{1}_{j}τ^{0}(_{j}φ_{4}^{(1)} + ζ_{j}φ_{4}^{(0)}) −

−_{j}a_{12}_{j}φ_{5}^{(0)} −_{j}a_{12}_{j}τ^{0}(_{j}φ_{5}^{(1)} + ζ_{j}φ_{5}^{(0)}) − −_{j}γ_{1} div (_{j}φ_{1}^{(0)},_{j}φ_{2}^{(0)},_{j}φ_{3}^{(0)})^{T} ,

jX̃_{5} = _{j}X̂_{5} −_{j}a_{2}jφ_{5}^{(0)}(x) −_{j}a_{2}_{j}τ^{0}(_{j}φ_{5}^{(1)} + ζ_{j}φ_{5}^{(0)}) −

−_{j}a_{12}_{j}φ_{4}^{(0)} −_{j}a_{12}_{j}τ^{0}(_{j}φ_{4}^{(1)} + ζ_{j}φ_{4}^{(0)}) − −_{j}γ_{2} div (_{j}φ_{1}^{(0)},_{j}φ_{2}^{(0)},_{j}φ_{3}^{(0)})^{T} ,

F̃ = F̂ −_{1}γ_{1}_{1}τ^{1}n_{0}(y)_{1}φ_{4}^{(0)} +_{2}γ_{1}_{2}τ^{1}n_{0}(y)_{2}φ_{4}^{(0)} −

−_{1}γ_{2}_{1}τ^{1}n_{0}(y)_{1}φ_{5}^{(0)} +_{2}γ_{2}_{2}τ^{1}n_{0}(y)_{2}φ_{5}^{(0)},
jF̃ = _{j}F̂ −_{j}γ_{1}_{j}τ^{1}n_{0}(y)_{j}φ_{4}^{(0)} −_{j}γ_{2}_{j}τ^{1}n_{0}(y)_{j}φ_{5}^{(0)}, j = 1, 2,
n_{0}(y) = (n(y), 0, 0)^{T}
|D_{x}^{β}V̂(x, ζ)| ≤ \frac{const}{1 + |x|^{1+|β|}} , |β| = \overline{0, 2},

β = (β_{1}, β_{2}, β_{3}) is the multi-index.

Now we introduce the following Green matrix-functions.

Let_{j}G_{τ} (x, y; ζ,_{j}ℝ^{3}) be the Green tensor (5×5 matrix) for the differential operator _{j}L(\frac{∂}{∂x}, ζ), the infinite homogeneous domain _{j}ℝ^{3} = ℝ^{3}\(\overline{D}_{j1} ∪\overline{D}_{j2}) and mixed boundary conditions (9), (10) on S_{j1} ∪ S_{j2}, respectively.

We have

jG_{τ} (x, y; ζ,_{j}ℝ^{3}) = _{j}Φ_{τ} (x − y, ζ) −_{j}g_{τ} (x, y; ζ,_{j}ℝ^{3}),

where _{j}Φ_{τ} (x − y, ζ) is a matrix of fundamental solutions of the operator jL(\frac{∂}{∂x}, ζ) which is constructed explicitly in terms of elementary functions [4, 7], and _{j}g_{τ} (x, y; ζ,_{j}ℝ^{3}) are regular matrices.

Let V̂(x, ζ) be a regular solution of Problem Â(ζ). Applying the Green matrices_{j}G_{τ}, the formula of general representation of a regular vector takes, in view of the contact and boundary conditions, the form

2χ_{D_{1}} (x)V̂(x, ζ) = ∫_{S} _{1}G_{τ} (x, z; ζ,_{1}ℝ^{3})[_{1}R_{τ} (\frac{∂}{∂z} , ζ, n)V̂(z, ζ)]^{+}d_{z}S −

− ∫_{S} [_{1}R̃_{τ}_{1}G_{τ}^{T} (x, z; ζ,_{1}ℝ^{3})]^{T} V̂^{+}(z, ζ)d_{z}S −

− ∫_{D_{1}} _{1}G_{τ} (x, z; ζ,_{1}ℝ^{3})_{1}H̃(z, ζ)dz −

− ∫_{S_{11}} [_{1}R̃_{τ}_{1}G_{τ}^{T} (x, z; ζ,_{1}ℝ^{3})]^{T} f̂(x, ζ)d_{z}S +

+ ∫_{S_{12}} _{1}G_{τ} (x, z; ζ,_{1}ℝ^{3})_{1}F̃(z, ζ)d_{z}S, x ∈ _{1}ℝ^{3}\S;  (11)

2χ_{D_{2}} (x)V̂(x, ζ) = − ∫_{S} _{2}G_{τ} (x, z; ζ,_{2}ℝ^{3})[_{1}R_{τ} (\frac{∂}{∂z} , ζ, n)V (z, ζ)]^{+}d_{z}S +

+ ∫_{S} [_{2}R̃_{τ}_{2}G_{τ}^{T} (x, z; ζ,_{2}ℝ^{3})]^{T} V̂^{+}(z, ζ)d_{z}S −

− ∫_{D_{2}} _{2}G_{τ} (x, z; ζ,_{2}ℝ^{3})_{2}H̃(z, ζ)dz +

+ ∫_{S} _{2}G_{τ} (x, z; ζ,_{2}ℝ^{3})F̃(z, ζ)d_{z}S −

− ∫_{S} [_{2}R̃_{τ} (\frac{∂}{∂z} , ζ, n)_{2}G_{τ}^{T} (x, z; ζ,_{2}ℝ^{3})]^{T} f̂(x, z)d_{z}S −

− ∫_{S_{21}} [_{2}R̃_{τ}_{2}G_{τ}^{T} (x, z; ζ,_{2}ℝ^{3})]^{T} f̂(x, ζ)d_{z}S +

+ ∫_{S_{22}} _{2}G_{τ} (x, z; ζ,_{2}ℝ^{3})_{2}F̃(z, ζ)d_{z}S, x ∈ _{2}ℝ^{3}\S,  (12)

where χ_{D_{j}} (x) is the characteristic function of the domain D_{j}, equal to 1 for x ∈ D_{j} and to 0 for x\overline{∈}D_{j}, j = 1, 2.

It is easy to check that if the vectors V̂^{+} and (_{1}R_{τ} V̂)^{+} are found by (11) for x ∈ D_{2} and by (11) for x ∈ D_{1}, then the substitution of these values in (11) for x ∈ D_{1} and in (12) for x ∈ D_{2} will give a solution of Problem Â(ζ).

To this end we introduce the following matrices:

jΨ(x, z, ζ) =
= ∥ (_{j}R̃_{τ}_{j}G_{τ}^{T} (x, z; ζ,_{j}ℝ^{3}))^{T} _{5×5}, −_{j}G_{τ} (x, z; ζ,_{j}ℝ^{3}) _{5×5}∥_{5×10} , (13)
ψ(x, ζ) = ∥ψ_{k}∥_{10×1} = (V̂^{+}, (_{1}R_{τ} V̂)^{+})^{T} .

It is easy to notice that by virtue of (11) and (12) we shall have

∀x ∈ D_{2} : ∫_{S} _{1}Ψ(x, z, ζ)ψ(z, ζ)d_{z}S = _{1}Θ(x),  (14)
∀x ∈ D_{1} : ∫_{S} _{2}Ψ(x, z, ζ)ψ(z, ζ)d_{z}S = _{2}Θ(x),  (15)

where

1Θ(x) = − ∫_{D_{1}} _{1}G_{τ} (x, z; ζ,_{1}ℝ^{3})_{1}H̃(z, ζ)dz −

− ∫_{S_{11}} [_{1}R̃_{τ}_{1}G_{τ}^{T} (x, z; ζ_{1}, ℝ^{3})]^{T}_{1}f̂(z, ζ)d_{z}S +

+ ∫_{S_{12}} _{1}G_{τ} (x, z; ζ,_{1}ℝ^{3})_{1}F̃(z, ζ)d_{z}S,

2Θ(x) = ∫_{D_{2}} _{2}G_{τ} (x, z; ζ,_{2}ℝ^{3})_{2}H̃(z, ζ)dz −

− ∫_{S} _{2}G_{τ} F̃d_{z}S + ∫_{S} [_{2}R̃_{τ}_{2}G_{τ}^{T} ]^{T} f̂d_{z}S +

+ ∫_{S_{21}} [_{2}R̃_{τ}_{2}G_{τ}^{T} ]^{T}_{2}f̂d_{z}S − ∫_{S_{22}} _{2}G_{τ} (x, z; ζ,_{2}ℝ^{3})_{2}F̃d_{z}S

are given vector-functions.

Now we shall construct auxiliary surfaces in the manner as follows: S̃_{1} is the closed surface which lies strictly inside the domain D_{1} covering the surface S_{11} ∪ S_{12}; S̃_{2} is the closed surface lying strictly inside the domain D_{2} covering the surface S. Clearly,

S̃_{1} ∩ S = ∅, S̃_{2} ∩ S = ∅.

Let {_{j}x^{k}}_{k=1}^{∞}, j = 1, 2, be an everywhere dense countable set of points on the auxiliary surface S̃_{j}, j = 1, 2.

By (14) and (15) we have

∫_{S} _{1}Ψ(_{2}x^{k}, z, ζ)ψ(z, ζ)d_{z}S = _{1}Θ(_{2}x^{k}), k = \overline{1, ∞},  (16)

∫_{S} _{2}Ψ(_{1}x^{k}, z, ζ)ψ(z, ζ)d_{z}S = _{2}Θ(_{1}x^{k}), k = \overline{1, ∞},  (17)

Denote the rows of the matrix _{j}Ψ considered as ten-component vectors (columns) by _{j}Ψ^{1},_{j}Ψ^{2}, …,_{j}Ψ^{5} (1 × 10 matrices) and investigate an infinite countable set of vectors

{_{1}Ψ^{l}(_{2}x^{k}, z, ζ)}_{k=1, l=1}^{∞,5} ∪ {_{2}Ψ^{l}(_{1}x^{k}, z, ζ)}_{k=1, l=1}^{∞,5}.  (18)

Theorem 2. Set (18) is linearly independent and complete in the vector Hilbert space L_{2}(S), i.e., it forms a basis in this space.

For the proof of this theorem see [8].

Rewrite (18) as follows:

{ψ^{k}(z)}_{k=1}^{∞},  (19)

where

ψ^{k}(z) ≡ _{a_{k}}Ψ^{l_{k}}(_{b_{k}}x^{q_{k}}, z, ζ),

a_{k} = k − 2[\frac{k − 1}{2}], b_{k} = 2[\frac{k + 1}{2}] − k + 1,
l_{k} = [\frac{k + 1}{2}] − 5 [ \frac{\frac{k+1}{2} − 1}{5}] , q_{k} = [ \frac{\frac{k+1}{2} + 4}{5}] ,

[a] is the integer part of the number a.

Clearly, by virtue of (16) and (17) the scalar product

(ψ^{k}, \overline{ψ}) = ∫_{S} [ψ^{k}]^{T} ψ ds = (ψ, \overline{ψ}^{k})

is known for any k; namely,

∫_{S} [ψ^{k}]^{T} ψ ds = _{a_{k}}Θ_{l_{k}}(_{b_{k}}x^{q_{k}}), k = \overline{1, ∞}.

It is obvious that the complex-conjugate system

\{\overline{ψ}^{k}(z)\}_{k=1}^{∞}  (20)

is complete, too.

Now let us determine the coefficients α_{k}, k = \overline{1, N}, assuming that the norm reduces to minimum (in L_{2}(S) with respect to system (20):

min_{α_{k}} ∥ψ(z) − ∑_{k=1}^{N} α_{k}\overline{ψ}^{k}(z)∥_{L_{2}(S)}.

As is well known, for this it is necessary and sufficient that

(ψ(z) − ∑_{k=1}^{N} α_{k}\overline{ψ}^{k}(z), \overline{ψ}^{j}(z)) = 0, j = \overline{1, N}.

Hence we come to an algebraic system of equations

∑_{k=1}^{N} α_{k}(\overline{ψ}^{k}, \overline{ψ}^{j}) = (ψ, \overline{ψ}^{j}), j = \overline{1, N},

with the right-hand side known and Gram determinant differing from zero, which determines the coefficients α_{k}. Therefore by virtue of the property of the space L_{2}(S) we have

lim_{N→∞} ∥ψ(z) − ∑_{k=1}^{N} α_{k}\overline{ψ}^{k}(z)∥_{L_{2}(S)} = 0.  (21)

Let us introduce the notation

ψ^{N}(z) = ∑_{k=1}^{N} α_{k}\overline{ψ}^{k}(z),

N V̂^{+} = (ψ^{N}_{1}, ψ^{N}_{2}, …, ψ^{N}_{5}, )^{T} ≡ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{1}^{k}, \overline{ψ}_{2}^{k}, …, \overline{ψ}_{5}^{k}, )^{T} ,

N(_{1}R_{τ} V̂)^{+} = (ψ^{N}_{6}, ψ^{N}_{7}, …, ψ^{N}_{10}, )^{T} ≡ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{6}^{k}, \overline{ψ}_{7}^{k}, …, \overline{ψ}_{10}^{k}, )^{T} .

We have in the sense of the metric of the space L_{2}(S)

ψ(z) = lim_{N→∞} ψ^{N}(z), V̂^{+} = lim_{N→∞}_{N} V̂^{+},

(_{1}R_{τ} V̂)^{+} = lim_{N→∞}_{N}(_{1}R_{τ} V̂)^{+}.

By substituting the obtained approximate values in (11) for x ∈ D_{1} and in (12) for x ∈ D_{2} and denoting the substitution result by _{N} V̂(x, ζ) we have

∀x ∈ D_{1} : 2_{N} V̂(x, ζ) = ∫_{S} _{1}G_{τ} (x, z; ζ,_{1}ℝ^{3})[ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{6}^{k}, …, \overline{ψ}_{10}^{k})^{T} ] d_{z}S −

− ∫_{S} [_{1}R̃_{τ} _{1}G_{τ}^{T} (x, z; ζ,_{1}ℝ^{3})]^{T} ×

× [ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{1}^{k}, …, \overline{ψ}_{5}^{k})^{T} ] d_{z}S +_{1}Θ(x),  (22)

∀x ∈ D_{2} : 2_{N} V̂(x, ζ) =−∫_{S} _{2}G_{τ} (x, z; ζ,_{2}ℝ^{3})[ ∑_{k=1}^{N}α_{k}(\overline{ψ}_{6}^{k}, …, \overline{ψ}_{10}^{k})^{T} ] d_{z}S+

+ ∫_{S} [_{2}R̃_{τ} _{2}G_{τ}^{T} (x, z; ζ,_{2}ℝ^{3})]^{T} ×

× [ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{1}^{k}, …, \overline{ψ}_{5}^{k})^{T} ] d_{z}S −_{2}Θ(x).  (23)

Thus determining the difference between (11)–(22) and (12)–(23) and applying the Cauchy–Buniakovsky inequality we find by virtue of (21) that for any ε > 0 there is a positive number N(ε) such that for N > N(ε) we shall have

|V̂(x, ζ) −_{N} V̂(x, ζ)| < ε,

where x ∈ \overline{D}′ ⋐ D_{1} ∪D_{2} (a strictly internal subdomain), V̂(x, ζ) is an exact solution of the problem, i.e.,

V̂(x, ζ) = lim_{N→∞}_{N} V̂(x, ζ), x ∈ \overline{D}′

tends to the limit uniformly (with respect to the metric C) in \overline{D}′.

As for the convergence with respect to the metric L_{2}, one can easily verify that the relation

lim_{N→∞} ∥V̂(x, ζ) −_{N} V̂(x, ζ)∥_{L_{2}(D_{1}∪D_{2})} = 0

is valid.

This method can be generalized to other more complicated problems. The foregoing discussion clearly implies that the solution of Problem

A^{t} is constructed by means of the Green tensor _{j}G_{τ} (x, y; ζ,_{j}ℝ^{3}) whose approximate values of can be constructed explicitly.

We shall give an algorithm of constructing Green tensors.

Clearly, to construct the tensor_{j}G_{τ} it is sufficient to construct the matrix jg_{τ}. We have

jg_{τ} (x, x^{0}; ζ,_{j}ℝ^{3}) = ∥_{j} g^{(1)}_{τ} ,_{j} g^{(2)}_{τ} , …,_{j} g^{(5)}_{τ} ∥_{5×5},

where the column-vectors _{j} g^{(s)}_{τ} (x, x^{0}; ζ,_{j}ℝ^{3}), s = \overline{1, 5}, are a solution of the following problem: In the infinite domain _{j}ℝ^{3} ≡ ℝ^{3}\(\overline{D}_{j1} ∪ \overline{D}_{j2}) determine

the vector _{j} g^{(s)}_{τ} by the conditions (j = 1, 2)

∀x ∈ _{j}ℝ^{3} : _{j}L(\frac{∂}{∂x}, ζ)_{j} g^{(s)}_{τ} (x, x^{0}; ζ,_{j}ℝ^{3}) = 0 (x^{0} ∈ _{j}ℝ^{3}),
∀y ∈ S_{j1} : _{j} g^{(s)}_{τ} (y, x^{0}; ζ,_{j}ℝ^{3}) = _{j} Φ^{(s)}_{τ} (y − x^{0}, ζ),
∀y ∈ S_{j2} : [_{j}R_{τ} (\frac{∂}{∂y} , ζ, n)_{j} g^{(s)}_{τ} (y, x^{0}; ζ,_{j}ℝ^{3})]^{+} =
=_{j}R_{τ} (\frac{∂}{∂y} , ζ, n)_{j} Φ^{(s)}_{τ} (y − x^{0}, ζ),

where s ∈ {1, …, 5} is a fixed number.

Let us construct auxiliary domains and surfaces in the manner as follows: D̃_{jk}, k = 1, 2, is the domain wholly lying in D_{jk}, S̃_{jk} are the boundaries of

D̃_{jk}, and _{j}S̃ = ∪^{2}_{k=1}S̃_{jk}.

We introduce a 5 × 5 matrix

jM_{τ} (y − x, ζ) = ∥_{j}M^{(1)}_{τ} ,_{j}M^{(2)}_{τ} , …,_{j}M^{(5)}_{τ} ∥_{5×5},

where

jM_{τ} (y − x, ζ) = \begin{cases}_{j}Φ_{τ} (y − x, ζ), & y ∈ S_{j1}, \\ _{j}R_{τ} (\frac{∂}{∂y} , ζ, n)_{j}Φ_{τ} (y − x, ζ), & y ∈ S_{j2}, \end{cases}
x ∈ R^{3}.

Let {_{j}x^{k}}_{k=1}^{∞} be an everywhere dense countable set of points on an auxiliary surface _{j}S̃.

Theorem 3. A countable set of vectors

{_{j}M^{(l)}_{τ} (y −_{j}x^{k}, ζ)}_{k=1, l=1}^{∞, 5}, y ∈ ∪^{2}_{m=1}S_{jm}  (24)

is linearly independent and complete in the Hilbert space L_{2}( ∪^{2}_{m=1}S_{jm}).

For the proof see [3, pp. 57, 123].

Enumerate the elements of (24) as follows:

\begin{aligned}_{j}ψ^{k}(y) & = _{j} M^{(p_{k})}_{τ} (y −_{j}x^{[ \frac{k+4}{5} ]}, ζ), k = \overline{1, ∞}, \\ p_{k} & = k − 5[\frac{k − 1}{5}], \end{aligned}  (25)

and assume {_{j}φ^{k}(y)}_{k=1}^{∞} to be the system obtained from (25) by orthonor-

malization on ∪^{2}_{m=1}S_{jm} by the Schmidt mehod, i.e.,

jφ^{k}(y) = ∑_{s=1}^{k}_{j}a_{ks}_{j}ψ^{s}(y), y ∈ ∪^{2}_{m=1}S_{jm}, k = \overline{1, ∞},  (26)

where _{j}a_{ks} are the orthonormalization coefficients.

It is easy to verify that

jφ^{k}(y) = \begin{cases}∑_{s=1}^{k}_{j}a_{ks}_{j} Φ^{(p_{s})}(y −_{j}x^{[ \frac{s+4}{5} ]}, ζ), & y ∈ S_{j1}, \\ ∑_{s=1}^{k}_{j}a_{ks}_{j}R_{τ} (\frac{∂}{∂y} , ζ, n)_{j} Φ^{(p_{s})}(y −_{j}x^{[ \frac{s+4}{5} ]}, ζ), & y ∈ S_{j2}. \end{cases}  (27)

We introduce the notation

j Ω^{(s)}^{(1)}(z, x^{0}) = _{j} Φ^{(s)}_{τ} (z − x^{0}, ζ), z ∈ S_{j1},

j Ω^{(s)}^{(2)}(z, x^{0}) = _{j}R_{τ} (\frac{∂}{∂y} , ζ, n)_{j} Φ^{(s)}_{τ} (z − x^{0}, ζ), z ∈ S_{j2}.

Let

j Ω^{(s)}(z, x^{0}) = _{j} Ω^{(s)}^{(k)}(z, x^{0}), z ∈ S_{jk}, k = 1, 2.

Clearly, by the property of the matrixt of fundamental solutions we have

j Ω^{(s)}(·, x^{0}) ∈ C^{∞}(S_{j1} ∪ S_{j2}), x^{0} ∈ _{j}ℝ^{3}.

By decomposing_{j} Ω^{(s)}(z, x^{0}) in a Fourier series with respect to the complete orthonormal system of vectors (26) we obtain

j Ω^{(s)}(z, x^{0}) ≈ ∑_{k=1}^{∞}_{j} Ω^{(s)}_{k}(x^{0})_{j}φ^{k}(z), z ∈ ∪^{2}_{l=1}S_{jl},

where

j Ω^{(s)}_{k}(x^{0}) = ∫_{S_{j1}∪S_{j2}} [_{j} Ω^{(s)}_{k}(y, x^{0})]^{T}_{j}\overline{φ^{k}}(y)d_{y}S.

Therefore by the property of the space L_{2}( ∪^{2}_{l=1}S_{jl}) we have

lim_{N→∞} ∥_{j} Ω^{(s)}(z, x^{0}) − ∑_{k=1}^{N}_{j} Ω^{(s)}_{k}(x^{0})_{j}φ^{k}(z)∥_{L_{2}( ∪^{2}_{l=1}S_{jl})} = 0.

We introduce the vectors

j g^{(s)}_{τ}^{(N)}(x, x^{0}) = ∑_{k=1}^{N} ∑_{m=1}^{k}_{j} Ω^{(s)}_{k}(x^{0})_{j}a_{km} _{j} Φ^{(p_{m})}_{τ} (x −_{j}x^{[ \frac{m+4}{5} ]}, ζ),

x, x^{0} ∈ _{j}ℝ^{3}, s = \overline{1, 5}.

Clearly, all the coefficients are uniquely defined. Now it is easy to show that the desired approximate value of the Green tensor is written as

j G^{(s)}_{τ}^{(N)}(x, x^{0}; ζ,_{j}ℝ^{3}) = _{j} Φ^{(s)}_{τ} (x − x^{0}, ζ) −_{j} g^{(s)}_{τ}^{(N)}(x, x^{0}) =

= _{j} Φ^{(s)}_{τ} (x − x^{0}, ζ) − ∑_{k=1}^{N} ∑_{m=1}^{k}_{j} Ω^{(s)}_{k}(x^{0})_{j}a_{km} ×

×_{j} Φ^{(p_{m})}_{τ} (x −_{j}x^{[ \frac{m+4}{5} ]}, ζ),  (28)

where j = 1, 2, s = \overline{1, 5}, x, x^{0} ∈ ℝ^{3}, N > 0 is a natural number. For the mathematical substantiation of this fact see [3, pp. 122–125].

We would like to emphasize that for fixed N the Green tensor _{j}G_{τ}^{(N)} is given explicitly by a finite number of quadratures.

Finally, on substituting (28) in (22) and (23), we respectively have

∀x ∈ D_{1} : 2_{N,_{N}}V̂(x, ζ) =∫_{S}_{1}G_{τ}^{(_{N})}(x, z; ζ,_{1}ℝ^{3})[ ∑_{k=1}^{N}α_{k}(\overline{ψ}_{6}^{k}, …, \overline{ψ}_{10}^{k})^{T} ]d_{z}S−

− ∫_{S} [_{1}R̃_{τ}_{1}G_{τ}^{T_{(_{N})}}(x, z; ζ,_{1}ℝ^{3})]^{T} ×

× [ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{1}^{k}, …, \overline{ψ}_{5}^{k})^{T} ]d_{z}S +_{1,_{N}}Θ(x),  (29)

∀x ∈ D_{2} : 2_{N,_{N}}V̂(x, ζ) = − ∫_{S} _{1}G_{τ}^{(_{N})}(x, z; ζ,_{2}ℝ^{3}) ×

× [ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{6}^{k}, …, \overline{ψ}_{10}^{k})^{T} ]d_{z}S +

+ ∫_{S} [_{2}R̃_{τ}_{2}G_{τ}^{T_{(_{N})}}(x, z; ζ,_{2}ℝ^{3})]^{T} ×

× [ ∑_{k=1}^{N} α_{k}(\overline{ψ}_{1}^{k}, …, \overline{ψ}_{5}^{k})^{T} ]d_{z}S −_{2,_{N}}Θ(x).  (30)

Thus we obtain the following relation with respect to the metric of the space C in \overline{D}′ ⋐ D_{1} ∪ D_{2}:

V̂(x, ζ) = lim_{\begin{matrix}N→∞ \\ _{N}→∞ \end{matrix}}_{N,_{N}}V̂(x, ζ).  (31)

As for an estimate in the closed domain \overline{D}_{1} ∪ \overline{D}_{2}, these estimates hold in the space L_{2}(\overline{D}_{1} ∪ \overline{D}_{2}).

Remark. Fundamental differential operators of the theory developed here have the form

P_{τ(k)}(\frac{∂}{∂x}, \frac{∂}{∂t}, n(x))V (x, t) = (T (\frac{∂}{∂x}, n(x))v(x, t) −

− n(x) ∑_{l=1}^{2} γ_{l}(1 + τ^{1} \frac{∂}{∂t})v_{3+l},

− (δ_{1k} + δ_{2k})v_{4} + (δ_{3k} + δ_{0k})δ_{1} \frac{∂v_{4}}{∂n} ,

− (δ_{1k} + δ_{3k})v_{5} + (δ_{2k} + δ_{0k})δ_{2} \frac{∂v_{5}}{∂n} )^{T} ,

Q_{(k)}(\frac{∂}{∂x}, n(x))V (x, t) = (v, (δ_{1k} + δ_{2k})δ_{1} \frac{∂v_{4}}{∂n} + (δ_{3k} + δ_{0k})v_{4},

(δ_{1k} + δ_{3k})δ_{2} \frac{∂v_{5}}{∂n} + (δ_{2k} + δ_{0k})v_{5})^{T} ,

where k = \overline{0, 3}, R_{τ} ≡ P_{τ(0)}, δ_{jk} is the Kronecker symbol.

Let _{1}D_{k}, k = \overline{0, 7}, be the nonintersecting domain within S with the Liapunov boundaries _{1}S_{k} (accordingly, _{1}D_{k} ∩ _{1}D_{m} = ∅, k ≠ m); _{2}D_{k} k = \overline{0, 7}, be the nonintersecting finite domains outside S with the Liapunov boundaries _{2}S_{k} (accordingly, _{2}D_{k} ∩ _{2}D_{m} = ∅, k ≠ m); _{1}D be a finite

domain bounded by the surfaces S ∪ ∪^{7}_{k=0}_{1}S_{k}, _{2}D be an infinite domain

bounded by the surfaces S ∪ ∪^{7}_{k=0}_{2}S_{k}. Consider a nonhomogeneous infinite

domain _{1}D ∪ S ∪_{2}D ≡ ℝ^{3}\ ∪^{2}_{j=1} ∪ ∪^{7}_{k=0}_{j}\overline{D}_{k}, where S is the contacting surface.

The method described can be generalized as well to the case where contact conditions on S and boundary conditions on _{j}S_{k} have the form

∀y ∈ S, t ≥ 0 : lim_{_{1}D∋x→y∈S} _{1}Q_{(k_{0})}(\frac{∂}{∂x}, n(y))V (x, t) −
− lim_{_{2}D∋x→y∈S} _{2}Q_{(k_{0})}(\frac{∂}{∂x}, n(y))V (x, t) = f(y, t),

lim_{_{1}D∋x→y∈S} _{1}P_{τ(k_{0})}(\frac{∂}{∂x}, \frac{∂}{∂t}, n(y))V (x, t) −

− lim_{_{2}D∋x→y∈S} _{2}P_{τ(k_{0})}(\frac{∂}{∂x}, \frac{∂}{∂t}, n(y))V (x, t) = F (y, t),
∀y ∈ _{j}S_{k}, t ≥ 0 : lim_{_{j}D∋x→y∈_{j}S_{k}} _{j}Q_{(k)}V (x, t) = _{j}f_{(k)}(y, t),
k = \overline{0, 3}, j = 1, 2,
∀y ∈ _{j}S_{k}, t ≥ 0 : lim_{_{j}D∋x→y∈_{j}S_{k}} _{j}P_{τ(k)}V (x, t) = _{j}F_{(k)}(y, t),
k = \overline{4, 7}, j = 1, 2,

where k_{0} ∈ {0, 1, …, 7} is a fixed number.

References

1. 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 Publishing Company, Amsterdam–New York–Oxford, 1979; Russian original: Nauka, Moscow, 1976.

2. V.D. Kupradze and T.V. Burchuladze. Dynamical problems of the theory of elasticity and thermoelasticity. (Russian) Current Problems in Mathematics (Russian), v. 7, 163–294. Itogi Nauki i Tekhniki Akad. Nauk SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhnich. Inform., Moscow, 1975; English translation: J. Soviet Math. 7(1977), No. 3.

3. T.V. Burchuladze and T.G. Gegelia. The development of the method of potential in the elasticity theory. (Russian) Metsniereba, Tbilisi, 1985.

4. T. V. Burchuladze. Nonstationary problems of generalized elastothermodiffusion for nonhomogeneous media. (Russian) Georgian Math. J. 1(1994), No. 6, 575–586.

5. T.V. Burchuladze. Green formulas in the generalized theory of elastothermodiffusion and their applications. (Russian) Trudy Tbilis. Mat. Inst. Razmadze 90(1988), 25–33.

6. T.V. Burchuladze. Mathematical theory of boundary value problems of thermodiffusion in deformed solid elastic bodies. (Russian) Trudy Tbilis. Mat. Inst. Razmadze 65(1980), 5–23.

7. T. V. Burchuladze. To the theory of dynamic mathematical problems of generalized elastothermodiffusion. (Russian) Trudy Tbilis. Mat. Inst. Razmadze 100(1992), 10–38.

8. T. V. Burchuladze. Approximate solutions of boundary value problems of thermodiffusion of solid deformable elastic bodies. (Russian) Trudy Tbilis. Mat. Inst. 71(1982), 19–30.

(Received 28.11.1994)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Alexidze St., Tbilisi 380093

Republic of Georgia