GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 3, 1997, 243-258

THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

OF ELASTOTHERMODIFFUSION WITH MIXED BOUNDARY CONDITIONS

T. BURCHULADZE AND YU. BEZHUASHVILI

Abstract. We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [1–8], we prove theorems on the existence and uniqueness of the classical solutions of these problems.

1. Statement of the problem. Let ℝ^{3} be the three-dimensional Euclidean space and let x = (x_{1}, x_{2}, x_{3}), y = (y_{1}, y_{2}, y_{3}), …, be the points of this space. Let D_{k} ⊂ ℝ^{3} be a finite domain bounded by a closed surface S_{k} ∈ Λ_{2}(α), α > 0, k = 0, 1, …, m [1]; moreover, S_{k} ∩ S_{j} = ∅, k ≠ j = \overline{0, m}, S_{0} covers all the others which in their turn do not cover each other, \overline{D}_{k} = D_{k} ∪ S_{k}, S = ∪_{k=0}^{m}S_{k}, D^{+} = D_{0}\ ∪_{k=1}^{m} \overline{D}_{k}; that is, D^{+} is a finite (bounded) domain with the boundary S; D^{−} = ℝ^{3}\ ∪_{k=1}^{m} \overline{D}_{k} is an infinite (unbounded) bounded domain with the boundary S′ = ∪_{k=1}^{m}S_{k}.

The principal (nonstationary) dynamic system of differential equations of the conjugate theory of thermodiffusion of homogeneous isotropic elastic body has the form [2, 3]:

A(\frac{∂}{∂x})v(x, t) − γ_{1} grad v_{4}(x, t) − γ_{2} grad v_{5}(x_{2}, t) − ρ\frac{∂^{2}v}{∂t^{2}} = q(x, t),
δ_{1}Δv_{4}(x, t) − a_{1} \frac{∂v_{4}}{∂t} − a_{12} \frac{∂v_{5}}{∂t} − γ_{1} \frac{∂}{∂t} div v(x, t) = q_{4}(x, t),
δ_{2}Δv_{5}(x, t) − a_{2} \frac{∂v_{5}}{∂t} − a_{12} \frac{∂v_{4}}{∂t} − γ_{2} \frac{∂}{∂t} div v(x, t) = q_{5}(x, t),

(1)

243 where v = (v_{1}, v_{2}, v_{3})^{T} = ∥v_{k}∥_{k=1}^{3} is the displacement vector, v_{4} is the deviation of temperature, v_{5} is “the chemical potential” of the medium, t is the time, A(\frac{∂}{∂x}) = ∥μδ_{ik} +(λ+μ)\frac{∂^{2}}{∂x_{i}∂x_{k}} ∥_{3×3} is the matrix Lamé differential (static) operator, Δ is the three-dimensional Laplace operator, δ_{ik} is the Kronecker symbol, q = (q_{1}, q_{2}, q_{3})^{T} is the vector of mass forces, q_{4} is the heat source, q_{5} is the diffusing mass source; λ, μ, γ_{1}, γ_{2}, δ_{1}, δ_{2}, a_{1}, a_{2}, a_{12}, ρ are the known real elastic, thermal, and diffusion constants, respectively, satisfying natural restrictions [3];

1991 Mathematics Subject Classification. 73B30, 73C25.

Key words and phrases. Thermodiffusion, oscillation problem, mixed boundary conditions, singular integral equations.

1072-947X/97/0500-0243$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

μ > 0, 3λ + 2μ > 0, ρ > 0,
a_{k} > 0, δ_{k} > 0, k = 1, 2, a_{1}a_{2} − a_{12}^{2} > 0,

T as a superscript denotes transposition.

Let us introduce a five-component vector V (x, t) = (v_{1}, v_{2}, …, v_{5})^{T} = (v, v_{4}, v_{5})^{T} and investigate the system (1) in the two cases:

I. When V (x, t) is represented in the form p > 0, stationary oscillations with frequency V (x, t) = Re[e^{−ipt}U(x, p)];

II. When V (x, t) is the Laplace–Mellin integral

V (x, t) = \frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞}e^{rt}U(x, τ)dτ, τ = σ + iq,

σ > 0, the general dynamical case.

It is easy to see that in both cases the system (1) is reduced to the form (for U(x, ω) = (u_{1}, u_{2}, …, u_{5})^{T} = (u, u_{4}, u_{5})^{T})

A(\frac{∂}{∂x})u − γ_{1} grad u_{4} − γ_{2} grad u_{5} + ρω^{2}u = φ,
δ_{1}Δu_{4} + iωa_{1}u_{4} + iωa_{12}u_{5} + iωγ_{1} div u = φ_{4},
δ_{2}Δu_{5} + iωa_{2}u_{5} + iωa_{12}u_{4} + iωγ_{2} div u = φ_{5},

(2)

where ω = p > 0 in the first case and ω = iτ = −q + iσ, σ > 0, in the second one.

We rewrite (2) in matrix-vector form

B(\frac{∂}{∂x}, ω)U(x, ω) = Φ(x),  (3)

where B(\frac{∂}{∂x}, ω) is a 5 × 5 matrix differential operator:

B(\frac{∂}{∂x}, ω) = ∥B_{jk}(\frac{∂}{∂x}, ω)∥_{5×5},
B_{jk}(\frac{∂}{∂x}, ω) = (μΔ + ρω^{2})δ_{jk} + (λ + μ)\frac{∂^{2}}{∂x_{j}∂x_{k}} , j, k = \overline{1, 3},

B_{jk}(\frac{∂}{∂x}, ω) = iωγ_{j−3} \frac{∂}{∂x_{k}} , k = \overline{1, 3}, j = 4, 5,
B_{kj}(\frac{∂}{∂x}, ω) = −γ_{j−3} \frac{∂}{∂x_{k}} , k = \overline{1, 3}, j = 4, 5,
B_{jj}(\frac{∂}{∂x}, ω) = δ_{j−3}Δ + iωa_{j−3}, j = 4, 5, B_{45} = 4_{j4} = iωa_{12};

Φ = (φ, φ_{4}, φ_{5})^{T} is a given vector.

We shall consider the following boundary value problems.

Problem M^{+}(ω). In the domain D^{+} find a regular vector U =(u, u_{4}, u_{5})^{T} (U ∈ C^{1}(\overline{D}^{+}) ∩ C^{2}(D^{+})), the solution of the system (3), by the boundary conditions:

U^{+}(y) = F^{(k)}(y), y ∈ S_{k}, k = \overline{0, m_{1}},
[R(\frac{∂}{∂y} , n)U(y)]^{+} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{1} + 1, m_{2}},
[Q(\frac{∂}{∂y} , n)U(y)]^{+} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{2} + 1, m_{3}},
[P (\frac{∂}{∂y} , n)U(y)]^{+} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{3} + 1, m_{4}},
[N_{1}(\frac{∂}{∂y} , n)U(y)]^{+} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{4} + 1, m_{5}},
[Ω_{1}(\frac{∂}{∂y} , n)U(y)]^{+} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{5} + 1, m_{6}},
[L_{1}(\frac{∂}{∂y} , n)U(y)]^{+} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{6} + 1, m_{7}},
[I_{1}(\frac{∂}{∂y} , n)U(y)]^{+} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{7} + 1, m}.

Here m_{i}, i = \overline{1, 7}, are arbitrary natural numbers such that 0 ≤ m_{1} ≤ m_{2} ≤ ⋯ ≤ m_{7} ≤ m,

RU = (HU, \frac{∂u_{4}}{∂n} , \frac{∂u_{5}}{∂n} , )^{T} ,
QU = (U, \frac{∂u_{4}}{∂n} , \frac{∂u_{5}}{∂n} )^{T} , P U = (HU, u_{4}, u_{5})^{T} ,
N_{1}U = ((u · n), HU − n(HU · n), \frac{∂u_{4}}{∂n} , \frac{∂u_{5}}{∂n} )^{T} ,
Ω_{1}U = (u − n(u · n), (HU · n), u_{4}, u_{5})^{T} ,
L_{1}U = ((u · n), HU − n(HU · n), u_{4}, u_{5})^{T} ,

I_{1}U = (u − n(u · n), (HU · n), \frac{∂u_{4}}{∂n} , \frac{∂u_{5}}{∂n} )^{T};

HU = T u − γ_{1}nu_{4} − γ_{2}nu_{5} is the thermodiffusion stress vector, T u is the elastic stress vector; (u · n), HU − n(HU · n) are the normal components of the displacement vector and the thermodiffusion stress vectors, respectively; F^{(k)}(y) ∈ C^{1,β}(S_{k}), β > 0, k = \overline{0, m}, are the tangential components of the displacement vector and the thermodiffusion stress vector; Φ = (φ, φ_{4}, φ_{5})^{T} ∈ C^{0,α}(\overline{D}^{+}), α > 0, are the known vectors, and n = n(y) = (n_{1}(y), n_{2}(y), n_{3}(y))^{T} = (n_{1}, n_{2}, n_{3})^{T} is the unit normal vector at the point y ∈ S (outer with respect to D^{+}) or at the point y ∈ S′ (outer with respect to D^{−}) [1, 2].

Problem M^{−}(ω). In the infinite domain D^{−} find a regular vector u = (u, u_{4}, u_{5}), the solution of the system (3), satisfying both the boundary conditions:

U^{−}(y) = F^{(k)}(y), y ∈ S_{k}, k = \overline{1, m_{1}},
[R(\frac{∂}{∂y} , n)U(y)]^{−} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{1} + 1, m_{2}},
[Q(\frac{∂}{∂y} , n)U(y)]^{−} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{2} + 1, m_{3}},
[P (\frac{∂}{∂y} , n)U(y)]^{−} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{3} + 1, m_{4}},
[N_{1}(\frac{∂}{∂y} , n)U(y)]^{−} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{4} + 1, m_{5}},
[Ω_{1}(\frac{∂}{∂y} , n)U(y)]^{−} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{5} + 1, m_{6}},
[L_{1}(\frac{∂}{∂y} , n)U(y)]^{−} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{6} + 1, m_{7}},
[I_{1}(\frac{∂}{∂y} , n)U(y)]^{−} = F^{(k)}(y), y ∈ S_{k}, k = \overline{m_{7} + 1, m},

and the following asymptotic equalities at infinity [2, 3]: the radiation condition of the thermodiffusion theory:

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

for ω = p > 0 and

\begin{matrix}u^{(j)}(x) = o(r^{−1}), \frac{∂u^{(j)}(x)}{∂x_{k}} = O(r^{−2}), j = 1, 2, k = 1, 2, 3, \\ u_{j}(x) = o(r^{−1}), \frac{∂u_{j}(x)}{∂x_{k}} = O(r^{−2}), j = 4, 5, k = 1, 2, 3, \end{matrix} (5)

for ω = iτ (σ > 0). Here

U = (u^{(1)} + u^{(2)}, u_{4}, u_{5})^{T} rot u^{(1)} = 0, div u^{(2)} = 0,

∏_{k=1}^{3}(Δ + λ_{k}^{2})u^{(1)} = 0, (Δ + λ_{4}^{2})u^{(2)} = 0, ∏_{k=1}^{3}(Δ + λ_{k}^{2})u_{j} = 0, j = 4, 5,

r is the radius-vector of the point x and λ_{k}^{2}, k = \overline{1, 4}, are the characteristic parameters [3]; Φ ∈ C^{0,β}(\overline{D}^{−}) is the given finitary vector.

For our purpose it is convenient to rewrite the boundary conditions in the equivalent form. Therefore instead of the above-stated problems we shall consider the following equivalent problems [2, 5, 6].

Problem M̂^{+}(ω). Find in D^{+} a regular vector U = (u, u_{4}, u_{5})^{T}, the solution of the system (3), by the boundary conditions

U^{+}(y) = f^{(k)}(y), y ∈ S_{k}, k = \overline{0, m_{1}},
[R(\frac{∂}{∂y} , n)U(y)]^{+} = f^{(k)}(y), y ∈ S_{k}, k = \overline{m_{1} + 1, m_{2}},
[Q(\frac{∂}{∂y} , n)U(y)]^{+} = f^{(k)}(y), y ∈ S_{k}, k = \overline{m_{2} + 1, m_{3}},
[P (\frac{∂}{∂y} , n)U(y)]^{+} = f^{(k)}(y), y ∈ S_{k}, k = \overline{m_{3} + 1, m_{4}},
[N(\frac{∂}{∂y} , n)U(y)]^{+} = f^{(k)}(y), y ∈ S_{k}, k = \overline{m_{4} + 1, m_{5}},
[Ω(\frac{∂}{∂y} , n)U(y)]^{+} = f^{(k)}(y), y ∈ S_{k}, k = \overline{m_{5} + 1, m_{6}},
[L(\frac{∂}{∂y} , n)U(y)]^{+} = f^{(k)}(y), y ∈ S_{k}, k = \overline{m_{6} + 1, m_{7}},
[I(\frac{∂}{∂y} , n)U(y)]^{+} = f^{(k)}(y), y ∈ S_{k}, k = \overline{m_{7} + 1, m},

where f^{(k)}(y) = (f_{1}^{(k)}(y), f_{2}^{(k)}(y), f_{3}^{(k)}(y), f_{4}^{(k)}(y), f_{5}^{(k)}(y))^{T}, y ∈ S_{k}, k = \overline{0, m_{4}}, f^{(k)}(y) = (f_{1}^{(k)}(y), …, f_{6}^{(k)}(y))^{T}, y ∈ S_{k}, k = \overline{m_{4} + 1, m}, are the vectors written, obviously, in terms of the boundary data of the problem M^{+}(ω) [7]:

NU = \{2μD(u · n) − (HU − n(HU · n)), (u · n), \frac{∂u_{4}}{∂n} , \frac{∂u_{5}}{∂n} \}^{T} ,

ΩU = {u − n(u · n), (HU · n) + 2μ ∑_{j=1}^{3} D_{j}(u_{j} − n_{j}(u · n)), u_{4}, u_{5}}^{T} ,

LU = {2μD(u · n) − ((HU − n(HU · n)), (u · n), u_{4}, u_{5}}^{T} ,

IU = {u − n(u · n), (HU · n) +

+ 2μ ∑_{j=1}^{3} D_{j}(u_{j} − n_{j}(u · n)), \frac{∂u_{4}}{∂n} , \frac{∂u_{5}}{∂n} \}^{T} ,

D(\frac{∂}{∂y} , n) = grad −n\frac{∂}{∂n} , D^{j}(\frac{∂}{∂y} , n) = \frac{∂}{∂y_{j}} − n_{j} \frac{∂}{∂n}

(see [1, 2]).

We consider also the corresponding exterior problem M̂^{−}(ω). Denote the homogeneous problems by M̂_{0}^{±}(ω). Use will be made of the conjugate operators Ñ, Ω̃, … which, as is easily seen, can be obtained from the corresponding operators by substituting the parameters γ_{1} and γ_{2} by iωγ_{1} and iωγ_{2}, respectively.

It should be noted that in [2, 4] the theorems on the existence and uniqueness of the solution of the problems M^{±}(ω) have been proved for 0 ≤ m_{1} ≤ m_{2} ≤ m_{3} ≤ m (m_{4} = m_{5} = m_{6} = m_{7} = m).

2. Uniqueness Theorems.

Theorem 1. In the class of regular vector-functions the homogeneous problem M̂_{0}^{+}(iτ), σ > 0, has only trivial solution.

Proof. The following generalized Green’s formula is known [2]:

∫_{D^{+}} \{v^{T} [A(\frac{∂}{∂x})u − γ_{1} grad u_{4} − γ_{2} grad u_{5}] +

+E(v, u) − γ_{1}u_{4} div v − γ_{2}u_{5} div v}dx = ∫_{S} v^{T} HUds,  (6)

where U = (u, u_{4}, u_{5})^{T}, V = (v, v_{4}, v_{5})^{T} are the regular vectors,

E(v, u) = \frac{3λ + 2μ}{3} div v div u + \frac{μ}{2} ∑_{k≠q=1}^{3} (\frac{∂v_{k}}{∂x_{q}} + \frac{∂v_{q}}{∂x_{k}} )(\frac{∂u_{k}}{∂x_{q}} + \frac{∂u_{q}}{∂x_{k}} ) +

+ \frac{μ}{3} ∑_{k,q=1}^{3} (\frac{∂v_{k}}{∂x_{k}} − \frac{∂v_{q}}{∂x_{q}} )(\frac{∂u_{k}}{∂x_{k}} − \frac{∂u_{q}}{∂x_{q}} ).

Let U(x) be the solution of the problem M̂_{0}^{+}(iτ) and let \overline{U}(x) be a complex-conjugate vector. Then from (6) we easily obtain

∫_{D^{+}} \{ρτ^{2}|u|^{2} + E(u, \overline{u}) + a_{1}|u_{4}|^{2} + a_{12}(u_{4}\overline{u}_{5} + \overline{u}_{4}u_{5}) +

+ a_{2}|u_{5}|^{2} + \frac{1}{\overline{τ}} ∑_{k=1}^{2} δ_{k}| grad u_{3+k}|^{2}\}dx =

= ∫_{S} [\overline{u}^{T} HU + \frac{1}{\overline{τ}} ∑_{k=1}^{2} δ_{k}u_{3+k} \frac{∂\overline{u}_{3+k}}{∂n} ]ds.  (7)

Taking into account the homogeneity of the boundary conditions, the integral with respect to S in (7) vanishes. Moreover, E(u, \overline{u}) ≥ 0, Im E(u, \overline{u}) = 0, a_{1}|u_{4}|^{2} +a_{12}(u_{4}\overline{u}_{5} +\overline{u}_{4}u_{5})+a_{2}|u_{5}|^{2} = |\sqrt{a}_{1}u_{4} + \frac{a_{12}}{\sqrt{a_{1}}} u_{5}|^{2} + \frac{a_{1}a_{2}−a_{12}^{2}}{a_{1}} |u_{5}|^{2} ≥ 0. Separation of the real and imaginary parts in (7) results in

∫_{D^{+}} \{ρ(σ^{2} − q^{2})|u|^{2} + E(u, \overline{u}) + |\sqrt{a_{1}}u_{4} + \frac{a_{12}}{\sqrt{a_{1}}} u_{5}|^{2} +

+\frac{a_{1}a_{2} − a_{12}^{2}}{a_{1}} |u_{5}|^{2} + \frac{σ}{|τ|^{2}} ∑_{k=1}^{2} δ_{k}| grad u_{3+k}|^{2}\}dx = 0,

q ∫_{D^{+}} [2ρσ|u|^{2} + \frac{1}{|τ|^{2}} ∑_{k=1}^{2} δ_{k}| grad u_{3+k}|^{2}]dx = 0.

Consequently, u(x) ≡ 0, u_{4}(x) ≡ 0, u_{5}(x) ≡ 0, that is, U(x) ≡ 0, x ∈ D^{+}.

Theorem 2. The homogeneous problem M̂_{0}^{−}(iτ) has only trivial solution.

This theorem follows directly from formulas (6) and (7) which, owing to (5), are likewise valid for an infinite domain D^{−}.

Theorem 3. The homogeneous problem M̂_{0}^{−}(p), p > 0, has only trivial solution.

We give an outline of the proof here. In [2,3] for the solution of the homogeneous problem we have the following Green’s formula:

\frac{2}{ip} ∑_{k=1}^{2} δ_{k} ∫_{D^{+}} | grad u_{3+k}|^{2}dx = ∫_{S} \{[\overline{u}^{T} HU − uH\overline{U}] +

+\frac{1}{ip} ∑_{k=1}^{2} δ_{k}(u_{3+k} \frac{∂\overline{u}_{3+k}}{∂n} + \overline{u}_{3+k} \frac{∂u_{3+k}}{∂n} )\}ds  (8)

written for the domain D^{−} ∩ H(0, R), where H(0, R) is a sphere centered at the origin, of radius R. Owing to this formula the proof of the theorem is analogous to that for the case of thermoelasticity [1, 2]. We have (see [2, 3, 4])

Theorem 4. The homogeneous problem M̂_{0}^{+}(p), p > 0 has the discrete spectrum of real (positive) eigenvalues.

In the sequel we shall need the uniqueness theorem whose proof is obvious.

Theorem 5. Homogeneous boundary value problems

B(\frac{∂}{∂x}, iτ)U(x) = 0, x ∈ D_{k}, k = \overline{1, m},

have only trivial solutions when one of the homogeneous boundary conditions of the problem M̂_{0}^{+}(iτ) is satisfied on the boundary of S_{k}.

3. Existence theorems for the problems M̂^{±}(iτ), M̂^{±}(p), p > 0. Consider the problem M̂^{±}(iτ). Let D^{(m_{4})} = D_{0}\ ∪_{k=1}^{m_{4}} \overline{D}_{k}; denote by G(x, y, D^{(m_{4})}, iτ) the Green’s tensor of the boundary value problem M̂^{+}(iτ) for 0 ≤ m_{1} ≤ m_{2} ≤ m_{3} ≤ m_{4}. Represent it in the form

G(x, y) = G(x, y, D^{(m_{4})}, iτ) = Γ(x − y, iτ) − g(x, y, D^{(m_{4})}, iτ),

where Γ(x − y, iτ) is the matrix of fundamental solutions of the operator B(\frac{∂}{∂x}, iτ), and g(x, y) = g(x, y, D^{m_{4}}, iτ) is the regular (matrix) solution of the boundary value problem

B(\frac{∂}{∂x}, iτ)g(x, y) = 0, x, y ∈ D^{(m_{4})},
g^{+}(z, y) = Γ(z − y, iτ), z ∈ S_{k}, k = \overline{0, m_{1}},
[R(\frac{∂}{∂z} , n)g(z, y)]^{+} = R(\frac{∂}{∂z} , n)Γ(z − y, iτ), z ∈ S_{k}, k = \overline{m_{1} + 1, m_{2}},

[Q(\frac{∂}{∂z} , n)g(z, y)]^{+} = Q(\frac{∂}{∂z} , n)Γ(z − y, iτ), z ∈ S_{k}, k = \overline{m_{2} + 1, m_{3}},
[P (\frac{∂}{∂z} , n)g(z, y)]^{+} = P (\frac{∂}{∂z} , n)Γ(z − y, iτ), z ∈ S_{k}, k = \overline{m_{3} + 1, m_{4}}.

According to the results obtained in [4], this nonhomogeneous problem is always solvable. We seek the solution of the problem M̂^{+}(iτ) in the form

U(x) = −\frac{1}{2} ∫_{D^{+}} G(x, y)Φ(y)dy −

− \frac{1}{2} ∑_{k=0}^{m_{1}} ∫_{S_{k}} [R̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ \frac{1}{2} ∑_{k=m_{1}+1}^{m_{2}} ∫_{S_{k}} G(x, y)f(y)ds −

− \frac{1}{2} ∑_{k=m_{2}+1}^{m_{3}} ∫_{S_{k}} [P̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ \frac{1}{2} ∑_{k=m_{3}+1}^{m_{4}} ∫_{S_{k}} [Q(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ ∑_{k=m_{4}+1}^{m_{5}} ∫_{S_{k}} [Ω̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ(y)ds +

+ ∑_{k=m_{5}+1}^{m_{6}} ∫_{S_{k}} [N(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ(y)ds +

+ ∑_{k=m_{6}+1}^{m_{7}} ∫_{S_{k}} [Ĩ(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ(y)ds +

+ ∑_{k=m_{7}+1}^{m} ∫_{S_{k}} [L(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ(y)ds,  (9)

where f(y) = f^{(k)}(y), y ∈ S_{k}, k = \overline{0, m_{4}}, is a given vector; φ(y), y ∈ S_{k}, k = \overline{m_{4} + 1, m} is an unknown six-dimensional vector. We can easily see that the differential equation and the boundary conditions on S_{k}, k = \overline{0, m_{4}}, are automatically satisfied for an arbitrary φ(y) ∈ C^{1,β}(S_{k}), k = \overline{m_{4} + 1, m}. To satisfy the rest of the boundary conditions, we obtain the following system of singular integral equations with respect to φ(y):

a(z)φ(z) + ∑_{m_{4}+1}^{m} ∫_{S_{k}} K(z, y)φ(y)ds +

+ ∑_{k=m_{4}+1}^{m} ∫_{S_{k}} K^{(1)}(z, y)φ(y)ds = l(z),  (10)

where K(z, y) is the singular matrix,

K(z, y)= \begin{cases}N(\frac{∂}{∂z} , n)[Ω̃(\frac{∂}{∂y} , n)Γ^{T} (z − y), iτ)]^{T} , z, y ∈ ∪^{m_{5}}_{k=m_{4}+1}S_{k}, \\ Ω(\frac{∂}{∂z} , n)[N(\frac{∂}{∂y} , n)Γ^{T} (z − y), iτ)]^{T} , z, y ∈ ∪^{m_{6}}_{k=m_{5}+1}S_{k}, \\ L(\frac{∂}{∂z} , n)[Ĩ(\frac{∂}{∂y} , n)Γ^{T} (z − y), iτ)]^{T} , z, y ∈ ∪^{m_{7}}_{k=m_{6}+1}S_{k}, \\ I(\frac{∂}{∂z} , n)[L(\frac{∂}{∂y} , n)Γ^{T} (z − y), iτ)]^{T} , z, y ∈ ∪^{m}_{k=m_{7}+1}S_{k}, \\ 0,  otherwise, \end{cases} (11)

a(z)= \begin{cases}−1, & z ∈ ∪^{m_{5}}_{k=m_{4}+1}S_{k} ∪ ∪^{m_{7}}_{k=m_{6}+1}S_{k}, \\ 1, & z ∈ ∪^{m_{6}}_{k=m_{5}+1}S_{k} ∪ ∪^{m}_{k=m_{7}+1}S_{k}, \end{cases}  (12)

K^{(1)}(z, y), z, y ∈ ∪_{k=m_{4}+1}^{m}S_{k} is a completely definite continuous matrix, l(z) is the known vector given in terms of the data of the problem; there is no need to write out this vector explicitly.

The system of integral equations (10) is singular; singular integrals involved in (10) are understood in the sense of the Cauchy principal value. These systems have been studied in detail in [1]. The only difference is in the completely continuous operators which do not affect the general theory. The system (10) is of normal type with zero index. We outline the proof, for example, in the case where z, y ∈ ∪_{k=m_{4}+1}^{m_{5}}S_{k}. For (10) we have

−φ(z) + ∑_{k=m_{4}+1}^{m_{5}} ∫_{S_{k}} N(\frac{∂}{∂z} , n)[Ω̃(\frac{∂}{∂y} , n)Γ^{T} (z − y, iτ)]^{T} φ(y)ds +
+ T(φ) = l(z).  (13)

(Here T is the completely continuous operator.) Direct calculation shows that the 6 × 6 symbolic matrix has the form (see [1,2,4,5,7])

σ^{+}(z, θ) = ∥σ_{jk}^{+}∥_{6×6}, σ_{jj} = −1, j = \overline{1, 6}, σ_{j4} = σ_{j}, j = \overline{1, 3},
σ_{jk} = 0 in all the rest cases,

σ_{j} = 2πi[K(z)(α_{j2} sin θ − α_{j1} cos θ) +

+ ∑_{k=1}^{3}(α_{k2} sin θ − α_{k1} cos θ)D^{j}(\frac{∂}{∂z} , n)n_{k}], j = 1, 3,

where K(z) = Dn(z) = ∑_{j=1}^{3} D^{j}(z)n_{j}(z) is the mean curvature of the surface S_{k} at the point z ∈ S_{k}. Hence, det σ^{+}(z, θ) = 1, and the singular operator of the system (10) is also of normal type. The system (13) has zero index. This follows from the fact that for the principal minors Δ_{j}, j = \overline{1, 6}, of the symbolic matrix σ^{+}(z, θ) we have Δ_{j} = (−1)^{j}. Thus, the lower bounds of modules of these minors are positive (see Mikhlin [8], p. 192). Moreover, the classical theory of Fredholm integral equations extends entirely to such systems. Consequently, the system (13) is regular-solvable.

Let us now prove that the system (10) is uniquely solvable. Indeed, let φ^{(0)}(y) ∈ C^{1,β}(S_{k}), k = \overline{m_{4} + 1, m}, be a solution of the homogeneous system corresponding to (10). Let us consider the potential

U^{∘}(x) = ∑_{k=m_{4}+1}^{m_{5}} ∫_{S_{k}} [Ω̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ^{(0)}(y)ds +

+ ∑_{k=m_{5}+1}^{m_{6}} ∫_{S_{k}} [N(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ^{(0)}(y)ds +

+ ∑_{k=m_{6}+1}^{m_{7}} ∫_{S_{k}} [Ĩ(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ^{(0)}(y)ds +

+ ∑_{k=m_{7}+1}^{m} ∫_{S_{k}} [L(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ^{(0)}(y)ds.

It is clear that U^{∘}(x) is the regular solution of the homogeneous problem M̂_{0}^{+}(iτ). By Theorem 1 we have

U^{∘}(x) ≡ 0, x ∈ D^{+}.  (14)

Furthermore,

B(\frac{∂}{∂x}, iτ)U^{∘}(x) = 0, x ∈ ∪^{m}_{k=m_{4}+1}D_{k}.  (15)

Moreover, taking into account certain Liapounov–Tauber type theorems proven in [1,2], we can show that

lim_{x→z} Ω(\frac{∂}{∂x}, n)U^{∘}(x) = 0, x ∈ ∪^{m_{5}}_{k=m_{4}+1}D_{k}, z ∈ ∪^{m_{5}}_{k=m_{4}+1}S_{k},
lim_{x→z} N(\frac{∂}{∂x}, n)U^{∘}(x) = 0, x ∈ ∪^{m_{6}}_{k=m_{5}+1}D_{k}, z ∈ ∪^{m_{6}}_{k=m_{5}+1}S_{k},
lim_{x→z} I(\frac{∂}{∂x}, n)U^{∘}(x) = 0, x ∈ ∪^{m_{7}}_{k=m_{6}+1}D_{k}, z ∈ ∪^{m_{7}}_{k=m_{6}+1}S_{k},
lim_{x→z} L(\frac{∂}{∂x}, n)U^{∘}(x) = 0, x ∈ ∪^{m}_{k=m_{7}+1}D_{k}, z ∈ ∪^{m}_{k=m_{7}+1}S_{k}.

(16)

On account of (15), (16) and Theorem 5 we obtain

U^{∘}(x) ≡ 0, x ∈ x ∈ ∪^{m_{5}}_{k=m_{4}+1}D_{k}.  (17)

Using the jump formulas of thermodiffusion potential, from (14) and (17) we have

φ^{(0)}(z) ≡ 0, z ∈ ∪^{m_{5}}_{k=m_{4}+1}S_{k}.

Thus the following theorem is valid.

Theorem 6. The solution of the inhomogeneous problem M̂^{+}(iτ), σ > 0 does exist; it is unique and can be represented in the form of (9), where φ(y) is the solution of the integral equation (10) which is solvable for an arbitrary right-hand side.

Consider now the problem M̂^{−}(iτ).

Let D_{∞}^{(m_{4})} = ℝ^{3}\ ∪_{k=1}^{m_{4}} D_{k}, G(x, y) = G(x, y, D_{∞}^{(m_{4})}, iτ) be the Green’s tensor of the boundary value problem M̂^{−}(iτ) for 1 ≤ m_{1} ≤ m_{2} ≤ m_{3} ≤ m. The existence of such a tensor follows from the results of [2, 4].

We seek the solution in the form

U(x) = −\frac{1}{2} ∫_{D^{−}} G(x, y)Φ(y)dy −

− \frac{1}{2} ∑_{k=1}^{m_{1}} ∫_{S_{k}} [R̃(\frac{∂}{∂y} , n)G^{T} (x, y; D_{∞}^{(m_{4})}, iτ)]^{T} f(y)ds +

+ \frac{1}{2} ∑_{k=m_{1}+1}^{m_{2}} ∫_{S_{k}} G(x, y)f(y)ds −

− \frac{1}{2} ∑_{k=m_{2}+1}^{m_{3}} ∫_{S_{k}} [P̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ \frac{1}{2} ∑_{k=m_{3}+1}^{m_{4}} ∫_{S_{k}} [Q(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ ∑_{k=m_{4}+1}^{m_{5}} ∫_{S_{k}} [Ω̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} ψ(y)ds +

+ ∑_{k=m_{5}+1}^{m_{6}} ∫_{S_{k}} [N(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} ψ(y)ds +

+ ∑_{k=m_{6}+1}^{m_{7}} ∫_{S_{k}} [Ĩ(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} ψ(y)ds +

+ ∑_{k=m_{7}+1}^{m} ∫_{S_{k}} [L(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} ψ(y)ds,  (18)

With respect to ψ(y) we obtain the following system of singular integral equations:

a(z)ψ(z) + ∑_{k=m_{4}+1}^{m} ∫_{S_{k}} K(z, y)ψ(y)ds +

+ ∑_{k=m_{4}+1}^{m} ∫_{S_{k}} K^{(2)}(z, y)ψ(y)ds = l(z),  (19)

where z ∈ ∪_{k=m_{4}+1}^{m}S_{k}, K(z, y), a(z) are determined by the formulas (11) and (12), respectively. K^{(2)}(z, y) is a completely definite continuous matrix. In the same way as for the problem M̂^{+}(iτ) we can show that the system (19) is solvable for an arbitrary right-hand side (here use is made of Theorems 2 and 5).

Thus the theorem below is valid.

Theorem 7. The solution of the inhomogeneous problem M̂^{−}(iτ) does exist; it is unique and can be represented in the form of (18), where ψ(y) is the solution of the integral equation (19) which is solvable for an arbitrary right-hand side.

Consider now the problem M̂^{−}(p) in the particular case where m_{5} = m_{6} = m_{7} = m, 1 ≤ m_{1} ≤ m_{2} ≤ m_{3} ≤ m_{4} ≤ m.

We seek the solution in the form

U(x) = −\frac{1}{2} ∫_{D^{−}} G(x, y)Φ(y)dy −

− \frac{1}{2} ∑_{k=1}^{m_{1}} ∫_{S_{k}} [R̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ \frac{1}{2} ∑_{k=m_{1}+1}^{m_{2}} ∫_{S_{k}} G(x, y)f(y)ds −

− \frac{1}{2} ∑_{k=m_{2}+1}^{m_{3}} ∫_{S_{k}} [P̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ \frac{1}{2} ∑_{k=m_{3}+1}^{m_{4}} ∫_{S_{k}} [Q(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} f(y)ds +

+ ∑_{k=m_{4}+1}^{m} ∫_{S_{k}} [Ω̃(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} φ(y)ds +

+ ∑_{k=m_{4}+1}^{m} ∫_{S_{k}} [N(\frac{∂}{∂y} , n)G^{T} (x, y)]^{T} χ(y)ds,  (20)

where G(x, y) = G(x, y; D_{∞}^{(m_{4})}, p), f(y) = f^{(k)}(y), y ∈ S_{k}, k = \overline{1, m_{4}}, is the given vector, and φ(y), χ(y) are the unknown vectors. For the first vector we obtain the following system of singular integral equations:

φ(z) + ∑_{k=m_{4}+1}^{m} ∫_{S_{k}} N(\frac{∂}{∂z} , n)[Ω̃(\frac{∂}{∂y} , n)G^{T} (z, y, D_{∞}^{(m_{4})}, p)]^{T} φ(y)ds =

= l(z) − ∑_{k=m_{4}+1}^{m} N(\frac{∂}{∂z} , n) ∫_{S_{k}} [N(\frac{∂}{∂y} , n)G^{T} (z, y, D_{∞}^{(m_{4})}, p)]^{T} χ(y)ds,(21)

where l(z) is the known vector given in terms of the problem’s data. Consider the homogeneous integral equation corresponding to (21),

φ(z) + ∑_{k=m_{4}+1}^{m} ∫_{S_{k}} N(\frac{∂}{∂z} , n) ×

×[Ω̃(\frac{∂}{∂y} , n)G^{T} (z, y, D_{∞}^{(m_{4})}, p)]^{T} φ(y)ds = 0.  (22)

The theorem below is proved in just the same way as in [1, 2].

Theorem 8. The equality of the parameter p^{2} to one of the eigenfrequencies of the problem

\begin{matrix}(Δ + \frac{ρp^{2}}{μ} )u(x) = 0, div u(x) = 0, x ∈ ∪^{m}_{k=m_{4}+1}D_{k}, \\ (u(x) − n(u(z) · n))^{+} = 0, (n · \frac{∂u(z)}{∂n})^{+} = 0, z ∈ ∪^{m}_{k=m_{4}+1}S_{k} \end{matrix} (23)

is the necessary and sufficient condition for equation (22) to have the nontrivial solution.

If p^{2} is a ν-multiple eigenfrequency of the problem, then the integral equation (22) has ν linearly independent solutions, and they coincide with the boundary values N(\frac{∂}{∂x}, n) of the operator over the solutions of the problem (23).

It follows from Theorem 8 that if p^{2} is not an eigenfrequency of the problem (23), then the integral equation (21) is solvable due to Fredholm’s first theorem, and the solution is representable by the formula (20), where χ(y) ≡ 0. If, however, p^{2} coincides with one of the eigenfrequencies of the problem (23), then the necessary and sufficient condition for solvability of equation (21) has the form

∑_{i=m_{4}+1}^{m} ∫_{S_{i}} \{ ∑_{k=m_{4}+1}^{m} N(\frac{∂}{∂z} , n) ×

× ∫_{S_{k}} [N(\frac{∂}{∂y} , n)G^{T} (z, y)]^{T} χ(y)d_{y}S\}ψ^{i}(z)d_{z}S = B_{i}, i = \overline{1, ν}, (24)

where {ψ^{i}(z)}_{i=1}^{ν} is a complete system of solutions of the associated homogeneous equation corresponding to (22), B_{i} are the known constants, and χ(y) ∈ C^{1,β}(S_{k}), y ∈ S_{k}, k = \overline{m_{4} + 1, m} is still an unknown vector.

Choosing χ(y) appropriately, we can show [5, 6, 7] that the conditions (24) are fulfilled.

Thus the following theorem holds.

Theorem 9. The solution of the problem M̂^{−}(p) does exist for 1 ≤ m_{1} < m_{2} < m_{3} < m_{4} < m; it is unique and can be represented in the form of (20), where φ(y) is the solution of the integral equation (21). If p^{2} is not an eigenvalue of the problem (23), then the integral equation (21) is solvable due to Fredholm’s first theorem, and χ(y) ≡ 0. If, however, p^{2} is the eigenvalue of the problem (23), then (21) is solvable due to Fredholm’s third theorem.

Now we can construct the Green’s tensor of this mixed problem for the region ℝ^{3}\ ∪_{k=1}^{m_{5}} \overline{D_{k}}. By means of that tensor we can solve the problem M̂^{−}(p) in the case 1 ≤ m_{1} < ⋯ < m_{5} < m (m_{6} = m_{7} = m). Continuing this process, we can solve the problem M̂^{−}(p) in the general case,

1 ≤ m_{1} < ⋯ < m_{7} < m.

The problem M̂^{+}(ρ), ρ > 0, can be investigated in a way similar to [4,5].

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 series in applied mathematics and mechanics, v. 25, North-Holland Publishing Company, Amsterdam–New York–Oxford, 1979; Russian original: Nauka, Moscow, 1976.

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

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

4. T. V. Burchuladze, Boundary value problems of thermoelasticity with mixed boundary conditions. (Russian) Trudy Tbilis. Mat. Inst. Razmadze 44(1974), 10–29.

5. T. V. Burchuladze and Yu. A. Bezhuashvili, On three-dimensional dynamic problems of the conjugate theory of elastothermodiffusion. (Russian) Differentsial’nye Uravneniya 17(1981), No. 8, 1446–1455.

6. T. V. Burchuladze and Yu. A. Bezhuashvili, On some three-dimensio-nal boundary value problems of the conjugate theory of elastothermodiffusion. (Russian) Trudy Gruz. Polytech. Inst. 247(1982), No. 2, 12–19.

7. Yu. A. Bezhuashvili, Spatial boundary value problems of the associated theory of thermoelasticity with mixed boundary conditions. (Russian) Trudy Tbilis. Mat. Inst. Razmadze 52(1976), 5–19.

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

(Received 02.10.1995)

Authors’ addresses:

A. Razmadze Mathematical Institute Department of Mathematics (N 3) Georgian Academy of Sciences Georgian Technical University 1, M. Aleksidze St., Tbilisi 380093 75, M. Kostava St., Tbilisi 380015 Georgia Georgia