GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 2, 1996, 177-200

ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS AND EIGENFREQUENCIES OF OSCILLATION BOUNDARY VALUE PROBLEMS OF THE LINEAR THEORY OF ELASTIC MIXTURES

M. SVANADZE

Abstract. The asymptotic behavior of eigenoscillation and eigenvector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.

Introduction

The wide application of composite materials has stimulated an intensive investigation of mathematical models of elastic mixtures. Many interesting results of theoretical and applied nature, presented mainly in the monographs [1–3] and the papers [4–8], have been obtained of late for these models.

Problems of the existence of frequencies of eigenoscillations are studied in [9–11] for internal boundary value problems of the diffusion and shift models of the linear theory of elastic mixtures. It is proved that by the diffusion model eigenoscillations do not arise in some composites, while in other composites there is a discrete spectrum of eigenoscillation frequencies. By the shift model all internal problems of oscillation have a discrete spectrum of eigenoscillation frequencies.

Lorentz’s well-known postulate that “asymptotic distribution of eigenoscillation frequencies does not depend on a shape of the body but depends on its volume” was proved by Weyl [12, 13] for two- and three-dimensional membranes. The same formulas were obtained by Courant [14] by means

177 of the variational method and by Carleman [15] who used the properties of Green functions and Tauber type theorems. The formula of asymptotic behavior of eigenfunctions was derived in [15] for a three-dimensional membrane.

1991 Mathematics Subject Classification. 73C15, 73D99, 73K20, 35E05, 35J55.

Key words and phrases. Boundary value problems for elastic mixtures, asymptotic behavior of eigenfunctions and frequencies, fundamental solution, Green tensors.

1072-947X/96/0300-0177$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

Weyl [16] proved Lorentz’s postulate for an isotropic three-dimensional elastic body and developed the law of asymptotic distribution of eigenoscillation frequencies. Plejel [17] proved this law by generalizing Carleman’s method and obtained formulas for asymptotic behavior of eigenvector-func-tions and potential energy density. Niemeyer [18] proved the same formulas by a different method and obtained the best estimate of the second term of these formulas.

Using Plejel’s method Burchuladze [19–20] proved Lorentz’s postulate for isotropic, orthotropic, and anisotropic plane elastic bodies, while Dikhamindzhia [21] for two- and three-dimensional isotropic bodies in couple-stress elasticity. These authors derived asymptotic formulas for eigenoscillation and eigenvector-function frequencies.

Asymptotic formulas of eigenfrequencies and eigenfunctions for problems of electromagnetic oscillation were obtained by Müller and Niemeyer in [22, 23].

Using V. Avakumović’s method asymptotic formulas of eigenfrequencies and eigenfunctions were obtained in [24, 25] for the first boundary value problem of different elliptic systems.

In this paper, Plejel’s method is used to prove Lorentz’s postulate for internal homogeneous oscillation boundary value problems in the shift model of the linear theory of a mixture of two isotropic elastic materials, and asymptotic formulas are derived for eigenfrequencies and eigenvector-func-tions.

1. Formulation of Boundary Value Problems

Let the finite domain Ω of the three-dimensional Euclidean space R^{3} be bounded by the surface S, S ∈ L_{2}(ε), 0 < ε ≤ 1 [26], \overline{Ω} = Ω ∪ S. It will be assumed that Ω is filled with a mixture of two isotropic elastic materials [1–3, 7]. The scalar product of vectors f = (f_{1}, f_{2}, …, f_{k}) and φ = (φ_{1}, φ_{2}, …, φ_{k}) will be denoted by f · φ = ∑_{j=1}^{k} f_{j}\overline{φ}_{j}, where \overline{φ}_{j} is the complex-conjugate to the number φ_{j}, f × φ = ∥ψ_{ml}∥_{k×k}, ψ_{ml} = f_{m}φ_{l}, m, l = \overline{1, k}, |f| = (∑_{l=1}^{k} f_{l}^{2})^{1/2}. The product of the matrix A = ∥A_{ml}∥_{p×k} by f denotes the vector Af = (∑_{j=1}^{k} A_{1j}f_{j}, ∑_{k=1}^{k} A_{2j}f_{j}, …, ∑_{j=1}^{k} A_{pj}f_{j}); A^{T} is the transposition of the matrix A. The trace of the square matrix A = ∥A_{ml}∥_{k×k} will be denoted by Sp A = ∑_{j=1}^{k} A_{jj}.

In the absence of mass force, the system of stationary oscillation equations in the shift model of the linear theory of two-component elastic mixtures has the form [3, 5, 6, 8, 27]

a_{1}Δu′ + b_{1} grad div u′ + cΔu′′ + d grad div u′′−
−α(u′ − u′′) + ω^{2}ρ_{11}u′ − ω^{2}ρ_{12}u′′ = 0,
cΔu′ + d grad div u′ + a_{2}Δu′′ + b_{2} grad div u′′+
+α(u′ − u′′) − ω^{2}ρ_{12}u′ + ω^{2}ρ_{22}u′′ = 0,

(1.1)

where u′ = (u′_{1}, u′_{2}, u′_{3}), u′′ = (u′′_{1}, u′′_{2}, u′′_{3}) are partial displacements, Δ is the three-dimensional Laplace operator, ω is an oscillation frequency, ω > 0, α ≥ 0,

a_{j} = μ_{j} − λ_{5}, b_{j} = μ_{j} + λ_{j} + λ_{5} + \frac{(−1)^{j}ρ_{3−j}α_{2}}{ρ_{1} + ρ_{2}} ,

ρ_{jj} = ρ_{j} + ρ_{12}, j = 1, 2, c = μ_{3} + λ_{5},

d = μ_{3} + λ_{3} + λ_{5} − \frac{ρ_{1}α_{2}}{ρ_{1} + ρ_{2}} = μ_{3} + λ_{4} − λ_{5} + \frac{ρ_{2}α_{2}}{ρ_{1} + ρ_{2}} , α_{2} = λ_{3} − λ_{4},

μ_{1}, μ_{2}, μ_{3}, λ_{1}, λ_{2}, …, λ_{5} are elastic constants of the mixture [1, 7].

In the sequel it will be assumed that the following conditions are fulfilled [1, 7]:

\begin{matrix}μ_{1} > 0, μ_{1}μ_{2} > μ_{3}^{2}, λ_{1} − \frac{ρ_{2}α_{2}}{ρ_{1} + ρ_{2}} + \frac{2}{3} μ_{1} > 0, λ_{5} ≤ 0, \\ ρ_{11} > 0, ρ_{11}ρ_{22} > ρ_{12}^{2}, \\ (λ_{1} − \frac{ρ_{2}α_{2}}{ρ_{1} + ρ_{2}} + \frac{2}{3} μ_{1})(λ_{2} + \frac{ρ_{1}α_{2}}{ρ_{1} + ρ_{2}} + \frac{2}{3} μ_{2}) > \\ > (λ_{3} − \frac{ρ_{1}α_{2}}{ρ_{1} + ρ_{2}} + \frac{2}{3} μ_{3})^{2}. \end{matrix}  (1.2)

System (1.1) can be rewritten in the matrix form as

A(D_{x}, ω^{2})U(x) ≡ [A(D_{x}) + αE_{0} + ω^{2}E]U(x) = 0,  (1.3)

where

U = (U_{1}, U_{2}, …, U_{6}) = (u′, u′′), x = (x_{1}, x_{2}, x_{3}) ∈ Ω,

D_{x} = (\frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , \frac{∂}{∂x_{3}} ),

A(D_{x}) = ∥\begin{matrix}A^{(1)}(D_{x}) & A^{(2)}(D_{x}) \\ A^{(2)}(D_{x}) & A^{(3)}(D_{x}) \end{matrix}∥_{6×6} , A^{(j)}(D_{x}) = ∥A_{kl}^{(j)}(D_{x})∥_{3×3},

A_{kl}^{(1)}(D_{x}) = a_{1}Δδ_{kl} + b_{1} \frac{∂^{2}}{∂x_{k}∂x_{l}} , A_{kl}^{(2)}(D_{x}) = cΔδ_{kl} + d \frac{∂^{2}}{∂x_{k}∂x_{l}} ,

A_{kl}^{(3)}(D_{x}) = a_{2}Δδ_{kl} + b_{2} \frac{∂^{2}}{∂x_{k}∂x_{l}} , j, k, l = 1, 2, 3,

E_{0} = ∥\begin{matrix}−I & I \\ I & −I \end{matrix}∥_{6×6} , E = ∥\begin{matrix}ρ_{11}I & −ρ_{12}I \\ −ρ_{12}I & ρ_{22}I \end{matrix}∥_{6×6} , I = ∥δ_{kl}∥_{3×3},

δ_{kl} is the Kronecker symbol.

A vector-function U is called regular in the domain Ω if U_{j} ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}) (j = \overline{1, 6}).

The internal homogeneous oscillation boundary value problems of the linear theory of elastic mixtures are formulated as follows:

Problem (K) (K=I, II, II, IV). Find a vector U regular in Ω satisfying system (1.3) and the homogeneous boundary condition

B^{(K)}(D_{z}, n(z))U(z) ≡ lim_{Ω∋x→z∈S} B^{(K)}(D_{x}, n(z))U(x) = 0,

where

B^{(K)}(D_{x}, n(z)) = \begin{cases}I & for K=I, \\ P(D_{x}, n(z)) & for K=II, \\ P(D_{x}, n(z)) + σI & for K=III, \\ ∥\begin{matrix}I & I \\ P^{(1)} + P^{(2)} & P^{(2)} + P^{(3)} \end{matrix}∥_{6×6} , & for K=IV, \end{cases}

I = ∥δ_{kl}∥_{6×6}, σ > 0, P(D_{x}, n(z)) is the stress operator in the theory of elastic mixtures [1],

P(D_{x}, n(z)=∥\begin{matrix}P^{(1)} & P^{(2)} \\ P^{(2)} & P^{(3)} \end{matrix}∥_{6×6} , P^{(j)} =∥P_{kl}^{(j)}∥_{3×3},

P_{kl}^{(1)}(D_{x}, n(z))=(μ_{1}−λ_{5})δ_{kl} \frac{∂}{∂n} +(μ_{1}+λ_{5})n_{l} \frac{∂}{∂x_{k}} +(λ_{1}− \frac{ρ_{2}α_{2}}{ρ_{1}+ρ_{2}} )n_{k} \frac{∂}{∂x_{l}} ,

P_{kl}^{(2)}(D_{x}, n(z))=(μ_{3}+λ_{5})δ_{kl} \frac{∂}{∂n} +(μ_{3}−λ_{5})n_{l} \frac{∂}{∂x_{k}} +(λ_{3}− \frac{ρ_{1}α_{2}}{ρ_{1}+ρ_{2}} )n_{k} \frac{∂}{∂x_{l}} ,

P_{kl}^{(3)}(D_{x}, n(z))=(μ_{2}−λ_{5})δ_{kl} \frac{∂}{∂n} +(μ_{2}+λ_{5})n_{l} \frac{∂}{∂x_{k}} +(λ_{2}− \frac{ρ_{2}α_{2}}{ρ_{1}+ρ_{2}} )n_{k} \frac{∂}{∂x_{l}} ,

n=(n_{1}, n_{2}, n_{3}) is the unit normal vector at a point z ∈S.

The internal pseudooscillation boundary value problems of the linear theory of elastic mixtures are formulated as follows:

Problem (K)_{f}^{∗} (K=I, II, III, IV). Find a vector U regular in Ω satisfying the system of equations

A(D_{x}, −κ^{2})U(x) = 0, x ∈ Ω,

and the boundary condition

B^{(K)}(D_{z}, n(z))U(z) = f(z), z ∈ S,

where κ > 0, f is a given six-component vector.

We have

Lemma 1.1. If f ∈ C^{1,δ_{1}}, 0 < δ_{1} ≤ 1, then problem (K)_{f}^{∗} has a unique regular solution (K=I,II,III,IV).

One can easily prove the uniqueness of a regular solution of problem (K)_{f}^{∗} by the Green formula [1]

∫_{Ω}[U(x) · A(D_{x})U(x) + W (U, U)] dx = ∫_{S}U(z) · P(D_{z}, n(z))U(z) d_{z}S, (1.4)

where W (U, U) is the doubled density of potential energy in the theory of elastic mixtures [1]. By virtue of conditions (1.2) W (U, U) is a nonnegative function for an arbitrary regular vector U [1]. The existence of solutions is proved by the potential method and the theory of singular integral equations [1, 26].

We introduce the notation

\begin{matrix}a = a_{1} + b_{1}, b = a_{2} + b_{2}, c_{0} = c + d, d_{1} = ab − c_{0}^{2}, \\ d_{2} = a_{1}a_{2} − c^{2}, ρ_{0} = ρ_{11}ρ_{22} − ρ_{12}^{2}, ρ_{3} = ρ_{11} + ρ_{22} − 2ρ_{12}, \\ q_{1} = a + b + 2c_{0}, q_{2} = aρ_{22} + bρ_{11} + 2c_{0}ρ_{12}, \\ q_{3} = a_{1} + a_{2} + 2c, q_{4} = a_{1}ρ_{22} + a_{2}ρ_{11} + 2cρ_{12}. \end{matrix} (1.5)

On account of (1.2) we obtain by (1.5)

\begin{matrix}a_{j} > 0, a > 0, b > 0, d_{j} > 0, \\ ρ_{0} > 0, ρ_{3} > 0, q_{l} > 0, j = 1, 2, l = \overline{1, 4}. \end{matrix}  (1.6)

Remark 1.1. If α = 0 then (1.1) implies

(Δ + k_{1}^{2})(Δ + k_{2}^{2}) (\begin{matrix}div u′ \\ div u′′ \end{matrix}) = 0,

(Δ + k_{3}^{2})(Δ + k_{4}^{2}) (\begin{matrix}rot u′ \\ rot u′′ \end{matrix}) = 0,

(1.7)

where k_{1}^{2}, k_{2}^{2} and k_{3}^{2}, k_{4}^{2} are the roots of the square equations d_{1}ξ^{2} − ω^{2}q_{2}ξ + ω^{4}ρ_{0} = 0 and d_{2}ξ^{2} − ω^{2}q_{4}ξ + ω^{4}ρ_{0} = 0, respectively. By virtue of conditions (1.6) we have k_{j}^{2} > 0 (j = \overline{1, 4}). Assume that k_{j} > 0 (j = \overline{1, 4}). By (1.7) it is clear that k_{1}, k_{2} and k_{3}, k_{4} are the wave numbers of longitudinal and transverse waves, respectively [3]. Let

c_{j} = ωk_{j}^{−1}, j = \overline{1, 4}.  (1.8)

The constants c_{1} and c_{2} will be velocities of longitudinal waves, while c_{3} and c_{4} the velocities of transverse waves [3]. Clearly, c_{1}^{2}, c_{2}^{2} and c_{3}^{2}, c_{4}^{2} are the roots of the equations

ρ_{0}ξ^{2} − q_{2}ξ + d_{1} = 0,  (1.9)
ρ_{0}ξ^{2} − q_{4}ξ + d_{2} = 0.  (1.10)

Remark 1.2. As is well known [28], by the classical theory of elasticity, in an isotropic body one longitudinal wave propagates with the velocity v_{1} = \sqrt{(λ + 2μ)ρ^{−1}}, and two transverse waves with the equal velocities v_{2} = \sqrt{μρ^{−1}} (here λ, μ are the Lamé constants, ρ is the body density). By the shift model, in mixture of two isotropic elastic materials the number of waves increases twofold [3]. For α = 0 two longitudinal waves propagate in the mixture with the velocities c_{1} and c_{2}, and four transverse waves with the velocities c_{3} and c_{4} (two pairs of waves having equal velocities) [3].

2. Fundamental Solution Matrix and Some of Its Properties

The fundamental solution matrix of the equation A(D_{x}, −κ^{2})U(x) = 0 is constructed in [27] in terms of elementary functions. It has the form

\begin{aligned}Γ(x, −κ^{2}) & = ∥\begin{matrix}Γ^{(1)}(x, −κ^{2}) & Γ^{(2)}(x, −κ^{2}) \\ Γ^{(2)}(x, −κ^{2}) & Γ^{(3)}(x, −κ^{2}) \end{matrix}∥_{6×6} , \\ Γ^{(j)} & = ∥Γ_{kl}^{(j)}∥_{3×3}, j = 1, 2, 3, \end{aligned}  (2.1)

Γ_{kl}^{(1)}(x, −κ^{2}) =\{\frac{1}{d_{2}} (a_{2}Δ − β_{3})(Δ − κ_{1}^{2})(Δ − κ_{2}^{2})δ_{kl} +

+(r_{1}^{(1)}Δ^{2} + r_{2}^{(1)}Δ + r_{3}^{(1)}) \frac{∂^{2}}{∂x_{k}∂x_{l}} \}γ,

Γ_{kl}^{(2)}(x, −κ^{2}) =\{ − \frac{1}{d_{2}} (cΔ+β_{2})(Δ−κ_{1}^{2})(Δ−κ_{2}^{2})δ_{kl}+

+(r_{1}^{(2)}Δ^{2} + r_{2}^{(2)}Δ + r_{3}^{(2)}) \frac{∂^{2}}{∂x_{k}∂x_{l}} \}γ,

Γ_{kl}^{(3)}(x, −κ^{2}) =\{\frac{1}{d_{2}} (a_{1}Δ − β_{1})(Δ − κ_{1}^{2})(Δ − κ_{2}^{2})δ_{kl} +

+(r_{1}^{(3)}Δ^{2} + r_{2}^{(3)}Δ + r_{3}^{(3)}) \frac{∂^{2}}{∂x_{k}∂x_{l}} \}γ,

k, l = 1, 2, 3, (2.2) where

β_{1} = α + κ^{2}ρ_{11}, β_{2} = α + κ^{2}ρ_{12}, β_{3} = α + κ^{2}ρ_{22},
r_{1}^{(1)} = \frac{b}{d_{1}} − \frac{a_{2}}{d_{2}} , r_{1}^{(2)} = −\frac{c_{0}}{d_{1}} + \frac{c}{d_{2}} , r_{1}^{(3)} = \frac{a}{d_{1}} − \frac{a_{1}}{d_{2}} ,
r_{2}^{(1)} = \frac{1}{d_{1}d_{2}} [2β_{2}(a_{2}d − cb_{2}) + β_{3}(2a_{2}b_{1} − 2cd − b_{1}b_{2} − d^{2})],
r_{2}^{(2)} = \frac{1}{d_{1}d_{2}} [β_{1}(a_{1}d − b_{2}c) + β_{2}(a_{2}b_{1} − 2cd + ab_{2} − d^{2}) + β_{3}(a_{1}d − b_{1}c)],
r_{2}^{(3)} = \frac{1}{d_{1}d_{2}} [β_{1}(2a_{1}b_{2} − 2cd + b_{1}b_{2} − d^{2}) + 2β_{2}(a_{1}d − b_{1}c)],
r_{3}^{(1)} = −\frac{1}{d_{1}d_{2}} (b_{1}β_{3}^{2} + 2dβ_{2}β_{3} + b_{2}β_{2}^{2}),
r_{3}^{(2)} = −\frac{1}{d_{1}d_{2}} (b_{1}β_{2}β_{3} + b_{2}β_{1}β_{2} + dβ_{1}β_{3} + dβ_{2}^{2}),

r_{3}^{(3)} = −\frac{1}{d_{1}d_{2}} (bβ_{2}^{2} + 2dβ_{1}β_{2} + b_{2}β_{1}^{2}), γ = ∑_{j=1}^{4} η_{j}γ_{j},

η_{j} = ∏_{\begin{matrix}l=1 \\ l≠j \end{matrix}}^{4}(κ_{j}^{2} − κ_{l}^{2})^{−1}, γ_{j}(x, −κ^{2}) = −\frac{e^{−κ_{j}|x|}}{4π|x|} , j = \overline{1, 4},

κ_{1}^{2}, κ_{2}^{2} and κ_{3}^{2}, κ_{4}^{2} are, respectively, the roots of the following square equations

d_{1}ξ^{2} − (aβ_{3} + bβ_{1} + 2c_{0}β_{2})ξ + β_{1}β_{3} − β_{2}^{2} = 0

and

d_{2}ξ^{2} − (a_{1}β_{3} + a_{2}β_{1} + 2cβ_{2})ξ + β_{1}β_{3} − β_{2}^{2} = 0.

By virtue of (1.6) we have κ_{j}^{2} > 0 (j = \overline{1, 4}). Assume that κ_{j} > 0 (j = \overline{1, 4}) and introduce the notation

\begin{aligned}q_{j}^{(0)} & =[d_{1}κ_{j}^{2}(κ_{j}^{2}−κ_{3−j}^{2})]^{−1}, q_{l}^{(0)}=[d_{2}κ_{l}^{2}(κ_{l}^{2}−κ_{7−l}^{2})]^{−1}, \\ q_{j}^{(1)} & =(bκ_{j}^{2}−β_{3})q_{j}^{(0)}, q_{j}^{(2)}=−(c_{0}κ_{j}^{2}+β_{2})q_{j}^{(0)}, q_{j}^{(3)}=(aκ_{j}^{2}−β_{1})q_{j}^{(0)}, \\ q_{l}^{(1)} & =(a_{2}κ_{j}^{2}−β_{3}) q_{l}^{(0)}, q_{l}^{(2)}=−(cκ_{l}^{2}+β_{2}) q_{l}^{(0)}, q_{l}^{(3)}=(a_{1}κ_{l}^{2}−β_{1}) q_{l}^{(0)}, \\ p_{k} & =ρ_{11}q_{k}^{(1)}−2ρ_{12}q_{k}^{(2)}+ρ_{22}q_{k}^{(3)}, j =1, 2, l =3, 4, k =\overline{1, 4}. \end{aligned}  (2.3)

We have

Lemma 2.1. The matrix Γ(x, −κ^{2}) has the following properties:

(a) Γ^{T} (x, −κ^{2}) = Γ(x, −κ^{2});

(b) Γ(−x, −κ^{2}) = Γ(x, −κ^{2});

(c) A(D_{x}, −κ^{2})Γ(x, −κ^{2}) = δ(x)I, where δ(x) is the Dirac distribution function;

(d) for x ≠ 0 the matrices Γ^{(1)}, Γ^{(2)}, Γ^{(3)} have the form

Γ^{(k)}(x, −κ^{2}) = grad div ∑_{j=1}^{2} q_{j}^{(k)}γ_{j}(x, −κ^{2}) −

− rot rot ∑_{l=3}^{4} q_{l}^{(k)}γ_{l}(x, −κ^{2}), k = 1, 2, 3;  (2.4)

(e) lim_{x→0} Sp \{[Γ(x, −κ^{2}) − Γ(x, −κ_{0}^{2})]E\} = \frac{1}{4π} κM + O(\sqrt{κ}), (2.5)

κ > κ_{0} > 0, κ → ∞;

(f) |\frac{∂^{s}}{∂x_{1}^{s_{1}}∂x_{2}^{s_{2}}∂x_{3}^{s_{3}}} Γ_{kl}(x, −κ^{2})| < \frac{c^{∗}e^{−α_{0}κ|x|}}{|x|^{1+s}},

where c^{∗} = const > 0, s = s_{1} + s_{2} + s_{3}, s_{1}, s_{2}, s_{3} are nonnegative integer numbers, α_{0} is a positive number not depending on κ and x,

M = c_{1}^{−3} + c_{2}^{−3} + 2(c_{3}^{−3} + c_{4}^{−3}).  (2.6)

Proof. The validity of properties (a), (b), (c) is proved immediately by verification.

Let us prove property (d). Taking into account the relations

Δγ_{j}(x, −κ^{2}) = κ_{j}^{2}γ_{j}(x, −κ^{2}),
Iγ_{j}(x, −κ^{2}) = \frac{1}{κ_{j}^{2}} ( grad div − rot rot )γ_{j}(x, −κ^{2}), x ≠ 0, j = \overline{1, 4},

we find by (2.2) that

Γ^{(1)}(x, −κ^{2}) = ∑_{j=1}^{4} η_{j}\{[\frac{1}{d_{2}κ_{j}^{2}} (a_{2}κ_{j}^{2} − β_{3})(κ_{j}^{2} − κ_{1}^{2})(κ_{j}^{2} − κ_{2}^{2}) +

+r_{1}^{(1)}κ_{j}^{4} + r_{2}^{(1)}κ_{j}^{2} + r_{3}^{(1)}] grad div −

−\frac{1}{d_{2}κ_{j}^{2}} (a_{2}κ_{j}^{2} − β_{3})(κ_{j}^{2} − κ_{1}^{2})(κ_{j}^{2} − κ_{2}^{2}) rot rot \}γ_{j}(x, −κ^{2}). (2.7)

Using the relations

\frac{1}{d_{2}κ_{j}^{2}} (a_{2}κ_{j}^{2} − β_{3})(κ_{j}^{2} − κ_{1}^{2})(κ_{j}^{2} − κ_{2}^{2}) + r_{1}^{(1)}κ_{j}^{4} + r_{2}^{(1)}κ_{j}^{2} + r_{3}^{(1)} =

= \frac{1}{d_{1}κ_{j}^{2}} (bκ_{j}^{2} − β_{3})(κ_{j}^{2} − κ_{3}^{2})(κ_{j}^{2} − κ_{4}^{2}), j = \overline{1, 4},

which are easy to verify, from (2.7) we obtain

Γ^{(1)}(x, −κ^{2}) = ∑_{j=1}^{4} η_{j}[\frac{1}{d_{1}κ_{j}^{2}} (bκ_{j}^{2} − β_{3})(κ_{j}^{2} − κ_{3}^{2})(κ_{j}^{2} − κ_{4}^{2}) grad div −

− \frac{1}{d_{2}κ_{j}^{2}} (a_{2}κ_{j}^{2} − β_{3})(κ_{j}^{2} − κ_{1}^{2})(κ_{j}^{2} − κ_{2}^{2}) rot rot ]γ_{j}(x, −κ^{2}) =

= ∑_{j=1}^{2} \frac{1}{d_{1}κ_{j}^{2}} (bκ_{j}^{2} − β_{3})(κ_{j}^{2} − κ_{3}^{2})(κ_{j}^{2} − κ_{4}^{2})η_{j} grad div γ_{j}(x, −κ^{2}) −

− ∑_{l=3}^{4} \frac{1}{d_{2}κ_{j}^{2}} (a_{2}κ_{j}^{2} − β_{3})(κ_{l}^{2} − κ_{1}^{2})(κ_{l}^{2} − κ_{2}^{2})η_{l} rot rot γ_{l}(x, −κ^{2}) =

= grad div ∑_{j=1}^{2} q_{j}^{(1)}γ_{j}(x, −κ^{2}) − rot rot ∑_{l=3}^{4} q_{l}^{(1)}γ_{l}(x, −κ^{2}).

For l = 2, 3 the validity of (2.4) is proved similarly to the above.

We shall now prove property (d). Let Φ(x, −κ^{2}) = Γ(x, −κ^{2})E. By (2.1), (2.2) the matrix Φ has the form

Φ = ∥\begin{matrix}Φ^{(1)} & Φ^{(2)} \\ Φ^{(3)} & Φ^{(4)} \end{matrix}∥_{6×6} , Φ^{(l)} = ∥Φ_{kj}^{(l)}∥_{3×3}, l = \overline{1, 4},
Φ^{(1)} = ρ_{11}Γ^{(1)} − ρ_{12}Γ^{(2)}, Φ^{(2)} = −ρ_{12}Γ^{(1)} + ρ_{22}Γ^{(2)},
Φ^{(3)} = ρ_{11}Γ^{(2)} − ρ_{12}Γ^{(3)}, Φ^{(4)} = −ρ_{12}Γ^{(2)} + ρ_{22}Γ^{(3)}.

Clearly, Sp Φ = Sp(ρ_{11}Γ^{(1)} − 2ρ_{12}Γ^{(2)} + ρ_{22}Γ^{(3)}). Hence on account of (2.3), (2.4) we obtain

Sp Φ(x, −κ^{2}) = Sp [ grad div ∑_{j=1}^{2} p_{j}γ_{j}(x, −κ^{2}) − rot rot ∑_{l=3}^{4} p_{l}γ_{l}(x, −κ^{2})] =

= Δ[ ∑_{j=1}^{2} p_{j}γ_{j}(x, −κ^{2}) + 2 ∑_{l=3}^{4} p_{l}γ_{l}(x, −κ^{2})] =

= ∑_{j=1}^{2} p_{j}κ_{j}^{2}γ_{j}(x, −κ^{2}) + 2 ∑_{l=3}^{4} p_{l}κ_{l}^{2}γ_{l}(x, −κ^{2}).  (2.8)

Using the equalities ∑_{j=1}^{2} p_{j}κ_{j}^{2} = q_{2}d_{1}^{−1}, ∑_{l=3}^{4} p_{l}κ_{l}^{2} = q_{4}d_{2}^{−1}, from (2.8) we have

Sp Φ(x, −κ^{2}) = −\frac{1}{4π|x|}[ ∑_{j=1}^{2} p_{j}κ_{j}^{2} + 2 ∑_{l=3}^{4} p_{l}κ_{l}^{2} − p_{5}(κ)|x|] +

+O(|x|) = −\frac{1}{4π|x|}[\frac{q_{2}}{d_{1}} + \frac{2q_{4}}{d_{2}} − p_{5}(κ)|x|] + O(|x|), |x| ≪ 1, (2.9)

where p_{5}(κ) = ∑_{j=1}^{2} p_{j}κ_{j}^{3} + 2 ∑_{l=3}^{4} p_{l}κ_{l}^{3}. The relation (2.9) readily implies

lim_{x→0} Sp [Φ(x, −κ^{2}) − Φ(x, −κ_{0}^{2})] = \frac{1}{4π} [p_{5}(κ) − p_{5}(κ_{0})], κ_{0} > 0. (2.10)

Let us now calculate the difference p_{5}(κ) − p_{5}(κ_{0}) for κ > κ_{0} > 0, κ → ∞. Assume that τ_{1}^{2}, τ_{2}^{2} and τ_{3}^{2}, τ_{4}^{2} are the roots of the square equations d_{1}ξ^{2} − κ^{2}q_{2}ξ + κ^{4}ρ_{0} = 0 and dξ^{2} − κ^{2}q_{4}ξ + κ^{4}ρ_{0} = 0, respectively. By (1.6) we have τ_{j}^{2} > 0 (j = \overline{1, 4}). Assuming that τ_{j} > 0 (j = \overline{1, 4}), we obtain τ_{j} = κc_{j}^{−1} (j = \overline{1, 4}), where c_{j} (j = \overline{1, 4}) are defined by (1.8). Taking into account the relations

τ_{1}^{2} + τ_{2}^{2} = κ^{2}q_{2}d_{1}^{−1}, τ_{1}^{2}τ_{2}^{2} = κ^{4}ρ_{0}d_{1}^{−1},
κ_{1}^{2} + κ_{1}κ_{2} + κ_{2}^{2} = τ_{1}^{2} + τ_{1}τ_{2} + τ_{2}^{2} + O(κ),
τ_{1} + τ_{2} = (κ_{1} + κ_{2})[1 + O(κ^{−1/2})], κ → ∞,

we find by (2.3) that

∑_{j=1}^{2} p_{j}κ_{j}^{3} = \frac{1}{κ_{1} + κ_{2}} [\frac{q_{2}}{d_{1}} (κ_{1}^{2} + κ_{1}κ_{2} + κ_{2}^{2}) − \frac{2κ^{2}ρ_{0}}{d_{1}} − \frac{αρ_{3}}{d_{1}} ] =

= \frac{1}{κ^{2}} \frac{τ_{1} + τ_{2}}{κ_{1} + κ_{2}} (τ_{1}^{3} + τ_{2}^{3}) + \frac{αρ_{3}}{d_{1}(κ_{1} + κ_{2})} =

= κ(c_{1}^{−3} + c_{2}^{−3}) + O(\sqrt{κ}), κ → ∞.

Similarly, one can show that ∑_{l=3}^{4} p_{l}κ_{l}^{2} = κ(c_{3}^{−3}+c_{4}^{−3})+O(\sqrt{x}), x → ∞. Therefore the equality

p_{5}(κ) − p_{5}(κ_{0}) = κM + O(\sqrt{κ}), κ > κ_{0} > 0, κ → ∞, (2.11)

is fulfilled. Putting (2.11) in (2.10) gives (2.5).

The validity of property (e) can be proved by Plejel’s method used to investigate fundamental solutions of oscillation equations of the classical theory of elasticity in [17].

3. Some Properties of Green Tensors

The matrix

G^{(K)}(x, y, −κ^{2}) = −Γ(x − y, −κ^{2}) + g^{(K)}(x, y, −κ^{2})

will be called the Green tensor of problem (K) (K=I, II, III, IV) if the

matrix g^{(K)} satisfies the homogeneous equation

A(D_{x}, −κ^{2}) g^{(K)}(z, y, −κ^{2}) = 0, x, y ∈ Ω,

and the boundary condition

B^{(K)}(D_{z}, n(z)) g^{(K)}(z, y, −κ^{2}) = B^{(K)}(D_{z}, n(z))Γ(z − y, −κ^{2}), z ∈ S, y ∈ Ω.

The matrix g^{(K)} (K=I, II, III, IV) is a solution of problem (K)_{f}^{∗} with

a special boundary value f = B^{(K)}Γ. The existence and uniqueness of g^{(K)} follows from Lemma 1.1.

Let u′ = (u′_{1}, u′_{2}, u′_{3}), u′′ = (u′′_{1}, u′′_{2}, u′′_{3}), Ũ = (u′, u′′). Assume that U, Ũ are the vector-functions with real components. Introduce the notation [1, 7]

ε′_{lj} = \frac{1}{2}(\frac{∂u′_{j}}{∂x_{l}} + \frac{∂u′_{l}}{∂x_{j}} ), ε′′_{lj} = \frac{1}{2}(\frac{∂u′′_{j}}{∂x_{l}} + \frac{∂u′′_{l}}{∂x_{j}} ),
ε′_{lj} = \frac{1}{2}(\frac{∂u′_{j}}{∂x_{l}} + \frac{∂u′_{l}}{∂x_{j}} ), ε′_{lj} = \frac{1}{2}(\frac{∂u′′_{j}}{∂x_{l}} + \frac{∂u′′_{l}}{∂x_{j}} ),
h_{lj} = \frac{1}{2}(\frac{∂u′_{j}}{∂x_{l}} − \frac{∂u′_{l}}{∂x_{j}} + \frac{∂u′′_{j}}{∂x_{l}} − \frac{∂u′′_{l}}{∂x_{j}} ),
h̃_{lj} = \frac{1}{2}(\frac{∂u′_{j}}{∂x_{l}} − \frac{∂u′_{l}}{∂x_{j}} + \frac{∂u′′_{j}}{∂x_{l}} − \frac{∂u′′_{l}}{∂x_{j}} ), l, j = 1, 2, 3,

W (U, Ũ) = ∑_{l,j=1}^{3} [(λ_{1}ε′_{ll} + λ_{3}ε′′_{ll})ε′_{jj} + 2μ_{1}ε′_{lj}ε′_{lj} + 2μ_{3}ε′′_{lj}ε′_{lj} +

+ (λ_{4}ε′_{ll} + λ_{2}ε′′_{ll})ε′′_{jj} + 2μ_{3}ε′_{lj}ε′′_{lj} + 2μ_{2}ε′′_{lj}ε′′_{lj} +

+ \frac{α_{2}}{ρ_{1} + ρ_{2}} (ρ_{2}ε′_{ll} + ρ_{1}ε′′_{ll})(ε′′_{jj} − ε′_{jj}) − 2λ_{5}h_{lj}h̃_{lj}].

Consider the functions

L[U, Ũ]=∫_{Ω} [W (U, Ũ)−αUE_{0}Ũ + κ^{2}UEŨ] dx,  (3.1)

L^{(K)}[U]=\begin{cases}∫_{Ω}[W (U, U)−αUE_{0}Ũ +κ^{2}UEŨ]dx≡L[U] & for K=I, \\ L[U]−2 ∫_{S} U(z) · PΓ_{j}(z − y, −κ^{2})d_{z}S & for K=II, \\ L^{(II)}[U]+σ ∫_{S}[U(z)−Γ_{j}(z − y, −κ^{2})]^{2}d_{z}S & for K=III, \\ L[U]−2 ∫_{S} u′′(z) · P̃Γ_{j}(z − y, −κ^{2})d_{z}S & for K=IV, \end{cases} (3.2)

where Γ_{j} is the jth column of the matrix Γ, j is some fixed number, j = \overline{1, 6}, P̃ = ∥P^{(1)} + P^{(2)}, P^{(2)} + P^{(3)}∥_{6×6}, σ > 0.

It is assumed that

R^{(K)} = \{U : U ∈ C^{2}(Ω) ∩ C^{1}(\overline{Ω}), B^{(K)}U(z) =
= B^{(K)}Γ_{j}(z − y, −κ^{2}), z ∈ S, y ∈ Ω}  (3.3)

is the set of admissible vector-functions for the functional L^{(K)} (K=I,II,III,IV). Let [17]

Γ̃(x − y, −κ^{2}) = \{1 − [1 − (\frac{r}{ρ_{y}(x)})^{m}]^{l}\}Γ(x − y, −κ^{2}),  (3.4)

where r = |x − y|, ρ_{y}(x) = max{r, ℓ_{y}}, ℓ_{y} is the distance from the point y to S, while m, l are natural numbers.

The following lemmas hold.

Lemma 3.1. The matrix Γ̃ has the properties:

a) Γ̃_{j} ∈ R^{(K)}, j = \overline{1, 6}, K=I,II,III,IV;

b) Γ̃_{j}(x − y, −κ^{2}) = Γ_{j}(x − y, −κ^{2}) for r ≥ ℓ_{y};

c) |\frac{∂^{s}}{∂x_{1}^{s_{1}}∂x_{2}^{s_{2}}∂x_{3}^{s_{3}}} Γ̃_{kj}(x − y, −κ^{2})| < \frac{const}{ℓ_{y}^{m}} e^{−α_{0}κr}r^{m−s−1} for r < ℓ_{y},

m ≥ s + 1, l ≥ s + 1, s = s_{1} + s_{2} + s_{3}, s_{1}, s_{2}, s_{3} are negative numbers, α_{0} is a positive number not depending on κ, x, y, and Γ̃_{j} is the jth column of the matrix Γ̃, j, k = \overline{1, 6}.

The validity of properties (a) and (b) immediately follows from (3.3), (3.4). Property (c) is easily proved by virtue of Lemma 2.1 and formula (3.4).

Lemma 3.2. If U and Ũ are vector-functions, regular in Ω, then the functionals L and L have the properties:

(a) L[U] ≥ 0. If L[U] = 0, then U(x) ≡ 0, x ∈ Ω;

(b) L[U, Ũ] = L[Ũ, U];

(c) L[U, U] = L[U];

(d) L[U + Ũ] = L[U] + 2L[U, Ũ] + L[Ũ];

(e) ∫_{Ω} Ũ · A(D_{x}, −κ^{2})U dx + L[U, Ũ] = ∫_{S} Ũ · PU dS; (3.5)

(f) L[Ũ, g_{j}^{(K)}] = ∫_{S} Ũ · P g_{j}^{(K)} dS, K=I,II,III,IV; (3.6)

(g) L[Ũ, g_{j}^{(I)}] = 0 for Ũ = U − g_{j}^{(I)}, U ∈ R^{(I)}, j = \overline{1, 6}.

Proof. The validity of properties (a), (b), (c), (d) is obvious by (3.1). Relation (3.5) follows from the Green formula [1]

∫_{Ω} [Ũ · A(D_{x})U + W (U, Ũ)] dx = ∫_{S} Ũ · PU dS.

If U = g_{j}^{(K)}, then from (3.5) we obtain equality (3.6). If U ∈ R^{(I)}, then

the vector Ũ = U − g_{j}^{(I)} satisfies the boundary condition Ũ(z) = 0, z ∈ S.

Therefore (3.6) implies L[Ũ, g_{j}^{(I)}] = 0.

Lemma 3.3. The functional L^{(K)} has a minimum value only on the vector

g_{j}^{(K)}, i.e.,

min_{U∈ R^{(K)}} L^{(K)}[U] = L^{(K)}[ g_{j}^{(K)}], K =I,II,III,IV.  (3.7)

Proof. If U ∈ R^{(I)}, Ũ ≡ U − g_{j}^{(I)}, then by Lemma 3.2 we have

L[U] = [L[g_{j}^{(I)} + Ũ] = L[g_{j}^{(I)}] + 2L[ g^{(I)}_{j}, Ũ] + L[Ũ] =

= L[g_{j}^{(I)}] + L[Ũ] ≥ L[g_{j}^{(I)}].

Let U ∈ R^{(III)}, g_{j} = g_{j}^{(III)}, Ũ = U − g_{j}. Then

L^{(III)}[U] =L[g_{j}] + 2L[g_{j}, Ũ]+L[Ũ]−2 ∫_{S} g_{j}PΓ_{j}dS−2 ∫_{S} Ũ · PΓ_{j}dS+
+σ ∫_{S}[g_{j} −Γ_{j}]^{2}dS + 2σ ∫_{S}(g_{j} −Γ_{j}) · ŨdS+σ ∫_{S} Ũ^{2}dS.  (3.8)

By virtue of (3.2), (3.6) and the boundary condition Pg_{j} + σg_{j} = PΓ_{j} + σΓ_{j} we find from (3.8) that

L^{(III)}[U] ≡ L^{(III)}[g_{j}] + L[Ũ] + σ ∫_{S} Ũ^{2}dS ≥ L^{(III)}[g_{j}].

Quite similarly, equality (3.7) is proved for K=II,IV. Property (a) of

Lemma 3.2 implies that the functional in the set R^{(K)} has a minimum only

on g_{j}^{(K)} (K=I,II,III,IV).

Lemma 3.4. The vector-functions g_{j}^{(I)}, g_{j}^{(II)}, g_{j}^{(III)}, g_{j}^{(IV)} satisfy the following relations:

(a) L^{(II)}[ g_{j}^{(III)}] ≤ L^{(III)}[ g_{j}^{(III)}], (3.9)

(b) g_{jj}^{(I)}(y, y, −κ^{2}) = −L[g_{j}^{(I)}]+

+ ∫_{S} Γ_{j}(z − y, −κ^{2}) · PΓ_{j}(z − y, −κ^{2}) d_{z}S; (3.10)

(c) g_{jj}^{(K)}(y, y, −κ^{2}) = − L^{(K)}[ g_{j}^{(K)}]−

− ∫_{S} Γ_{j}(z − y, −κ^{2}) · PΓ_{j}(z − y, −κ^{2}) d_{z}S, K=II,III; (3.11)

(d) g_{jj}^{(IV)}(y, y, −κ^{2})=− L^{(IV)}[ g_{j}^{(IV)}]+∫_{S}[Γ′_{j}P′_{j}−2Γ′′_{j}P′_{j}−Γ′′_{j}P′′_{j}] d_{z}S; (3.12)

(e) g_{jj}^{(I)}(y, y, −κ^{2}) = − L^{(III)}[g_{j}^{(I)}]−

− ∫_{S} Γ_{j}(z − y, −κ^{2}) · PΓ_{j}(z − y, −κ^{2}) d_{z}S; (3.13)

(f) g_{jj}^{(I)}(y, y, −κ^{2}) ≤ g_{jj}^{(III)}(y, y, −κ^{2}) ≤ g_{jj}^{(II)}(y, y, −κ^{2}), (3.14)

g_{jj}^{(I)}(y, y, −κ^{2}) ≤ g_{jj}^{(IV)}(y, y, −κ^{2}), (3.15)

where y ∈ Ω, Γ′_{j} = (Γ_{1j}, Γ_{2j}, Γ_{3j}), Γ′′_{j} = (Γ_{4j}, Γ_{5j}, Γ_{6j}), P′_{j} = (P^{(1)} + P^{(2)})Γ′_{j}, P′′_{j} = (P^{(2)} + P^{(3)})Γ′′_{j}.

Proof. The validity of property (a) is obvious by virtue of (3.2).

(b) It is easy to show that any regular vector U can be represented in the form

U(y) = ∫_{S} { G^{(K)}(y, z, −κ^{2})PU(z) − [P G^{(K)}(z, y, −κ^{2})]^{T} U(z)}d_{z}S −
− ∫_{Ω} G^{(K)}(y, z, −κ^{2})A(D_{z}, −κ^{2})U(z) dz,  (3.16)
κ > 0, y ∈ Ω, K=I,II,III,IV.

If U = g_{j}^{(I)}, then (3.16) implies

g_{jj}^{(I)}(y, y, −κ^{2}) = − ∫_{S} Γ_{j}(z − y, −κ^{2}) · P G^{(I)}_{j}(z, y, −κ^{2}) d_{z}S. (3.17)

On the other hand, by the Green formula (1.4) we have

∫_{Ω} U · A(D_{x}, −κ^{2})U dx + L[U] = ∫_{S} U · PU d_{z}S.  (3.18)

Hence for U = g_{j}^{(K)} we have

L[ g_{j}^{(K)}] = ∫_{S} g_{j}^{(K)}(z, y, −κ^{2}) · P g_{j}^{(K)}(z, y, −κ^{2}) d_{z}S  (3.19)

which on account of (3.17) gives

L[g_{j}^{(I)}] = ∫_{S} g_{j}^{(I)}(z, y, −κ^{2}) · (PG_{j}^{(I)} + PΓ_{j}) d_{z}S =

= −g_{jj}^{(I)} + ∫_{S} Γ_{j}(z − y, −κ^{2}) · PΓ_{j}(z − y, −κ^{2}) d_{z}S.

(c) Formula (3.16) implies for g_{j}^{(II)} that

g_{j}^{(II)}(y, y, −κ^{2}) = ∫_{S} G_{j}^{(II)}(y, z, −κ^{2}) · P g_{j}^{(II)}(z, y, −κ^{2}) d_{z}S =

= ∫_{S} G_{j}^{(II)}(y, z, −κ^{2}) · PΓ_{j}(z − y, −κ^{2}) d_{z}S.

By relation (3.19), from (3.2) we have

L^{(II)}[ g_{j}^{(II)}] = L[ g_{j}^{(II)}] − 2 ∫_{S} g_{j}^{(II)} · PΓ_{j} d_{z}S = ∫_{S} g_{j}^{(II)} · P g_{j}^{(II)} d_{z}S −

− 2 ∫_{S} g_{j}^{(II)} · PΓ_{j} d_{z}S = − ∫_{S} g_{j}^{(II)} · PΓ_{j} d_{z}S =

= − ∫_{S} (G_{j}^{(II)} + Γ_{j}) · PΓ_{j} d_{z}S = −g_{jj}^{(II)} − ∫_{S} Γ_{j} · PΓ_{j} d_{z}S.

The validity of formulas (3.12), (3.13), and (3.11) for K=III is proved similarly.

(f) Using Lemma 3.3 and relations (3.9) and (3.13), we find from (3.11) that

−g_{jj}^{(II)} ≤ L^{(II)}[ g_{j}^{(III)}] + ∫_{S} Γ_{j} · PΓ_{j} dS ≤ L^{(III)}[ g_{j}^{(III)}] + ∫_{S} Γ_{j} · PΓ_{j} dS =

= − g^{(III)} ≤ L^{(III)}[ g^{(I)}] + ∫_{S} Γ_{j} · PΓ_{j} dS = −g_{jj}^{(I)}.

By Lemma 3.3 and equalities (3.2) and (3.10), from (3.12) we have

−g_{jj}^{(IV)} ≤ L^{(IV)}[g_{j}^{(I)}] − ∫_{S} [Γ′_{j}P′_{j} − 2Γ′′_{j}P′_{j} − Γ′′_{j}P′′_{j}] dS =

= L[ g^{(I)}_{j}] − ∫_{S} [2Γ′′_{j}(P′_{j} + P′′_{j}) + Γ′_{j}P′_{j} − 2Γ′′_{j}P′_{j} − Γ′′_{j}P′′_{j}] dS =

= L[ g^{(I)}_{j}] − ∫_{S} (Γ′_{j}P′_{j} + 2Γ′′_{j}P′_{j}) dS = −g_{jj}^{(I)}.

Lemma 3.5. The function g_{jj}^{(K)} satisfies the relation

|g_{jj}^{(K)}(y, y, −κ^{2})| < \frac{const}{ℓ_{y}^{1+δ}}, y ∈ Ω, δ > 0, K=I,II,III,IV. (3.20)

Proof. First we prove the validity of the formula

∫_{B(y,ℓ_{y})} Γ̃_{j}(x − y, −κ^{2}) · A(D_{x}, −κ^{2})Γ̃_{j}(x − y, −κ^{2}) ≤ g_{jj}^{(I)}(y, y, −κ^{2}) dx ≤

≤ ∫_{S} Γ_{j}(z − y, −κ^{2}) · PΓ_{j}(z − y, −κ^{2}) d_{z}S,  (3.21)

where B(y, ℓ_{y}) is the ball with center y and radius ℓ_{y}. By property (a) of Lemma 3.2 relation (3.10) implies

g_{jj}^{(I)}(y, y, −κ^{2}) dx ≤ ∫_{S} Γ_{j}(z − y, −κ^{2}) · PΓ_{j}(z − y, −κ^{2}) d_{z}S. (3.22)

On the other hand, for U = Γ̃_{j} formula (3.18) gives

∫_{Ω} Γ̃_{j} · A(D_{x}, −κ^{2})Γ̃_{j} dx + L[Γ̃_{j}] = ∫_{S} Γ̃ · PΓ̃ dS.

Hence by Lemma 3.1 we have

L[Γ̃_{j}] = − ∫_{Ω} Γ̃_{j} · A(D_{x}, −κ^{2})Γ̃_{j} dx + ∫_{S} Γ_{j} · PΓ_{j} dS.  (3.23)

By Lemma 3.3 and relation (3.23), from (3.10) we obtain

g_{jj}^{(I)}(y, y, −κ^{2}) dx ≥ −L^{(I)}[Γ̃_{j}] + ∫_{Ω} Γ_{j} · PΓ_{j} dS =

= ∫_{B(y,ℓ_{y})} Γ̃_{j} · A(D_{x}, −κ^{2})Γ̃_{j} dx.  (3.24)

(3.22) and (3.24) imply that (3.21) is valid.

Due to Lemma 3.1 we have

|Γ̃_{mj}(x − y, −κ^{2})| ≤ const ·r^{4}ℓ_{y}^{−5} ≤ const ·ℓ_{y}^{−1},
|κ^{2}Γ̃_{mj}(x − y, −κ^{2})| ≤ const ·(xr)^{2}e^{−α_{0}κr}r^{2}ℓ_{y}^{−5} ≤
≤ const ·ℓ_{y}^{−3}, m, j = \overline{1, 6},
|Γ̃_{j}(x − y, −κ^{2}) · A(D_{x}, −κ^{2})Γ̃_{j}(x − y, −κ^{2})| ≤
≤ const ·r^{2}ℓ_{y}^{−6} ≤ const ·ℓ_{y}^{−4}.

(3.25)

With (3.25) taken into account, formula (3.21) readily implies

|g_{jj}^{(I)}(y, y, −κ^{2})| < \frac{const}{ℓ_{y}^{1+δ}}, y ∈ Ω, δ > 0.  (3.26)

The inequality

|g_{jj}^{(K)}(y, y, −κ^{2})| < \frac{const}{ℓ_{y}^{1+δ}}, y ∈ Ω, δ > 0, K=I,II,III,IV, (3.27)

is proved by the method which in [17] is used to estimate the regular part of the Green tensor from the second boundary value problem of oscillation in the classical theory of elasticity.

On account of inequalities (3.14), (3.15), (3.26), (3.27) we easily find that relation (3.20) is valid.

In a similar manner, as in the classical theory of elasticity [26], we prove

Lemma 3.6. The Green tensor G^{(K)} has the property

G^{(K)}^{T} (x, y, −κ^{2}) = G^{(K)}(y, x, −κ^{2}),

where x, y ∈ Ω, κ > 0, K=I,II,III,IV.

We introduce the notation

F = \frac{1}{\sqrt{ρ_{11} + ρ_{22} + 2\sqrt{ρ_{0}}}} ∥\begin{matrix}(ρ_{11} + \sqrt{ρ_{0}})I & −ρ_{12}I \\ −ρ_{12}I & (ρ_{22} + \sqrt{ρ_{0}})I \end{matrix}∥_{6×6} , G^{(K)} = F G^{(K)}F.

It is obvious that the equalities F^{2} = E and

G^{(K)}^{T} (x, y, −κ^{2}) = G^{(K)}(y, x, −κ^{2})  (3.28)

are fulfilled.

4. Asymptotic Behavior of Eigenoscillation and Eigenvector-Function Frequencies

Let U be a regular solution of problem (K) (K=I,II,III,IV). Then in (3.16) the integral through the surface S is zero. Taking into account the equality

A(D_{x}, −κ_{0}^{2})U(y) = −(ω^{2} + κ_{0}^{2})EU(y), κ_{0} > 0,

from (3.16) we obtain

U(x) = (ω^{2} + κ_{0}^{2}) ∫_{Ω} G^{(K)}(x, y, −κ_{0}^{2})EU(y) dy.  (4.1)

After multiplying (4.1) from the left by the matrix F, by (3.28) we obtain a system of second-kind Fredholm integral equation with symmetric kernel

V (x) = (λ^{∗} + κ_{0}^{2}) ∫_{Ω} G^{(K)}(x, y, κ_{0}^{2})V (y) dy,  (4.2)

where V = F U, λ^{∗} = ω^{2}.

One can easily verify [14] that system (4.2) has a discrete spectrum of

nonnegative eigenvalues. If ω_{1}^{(K)}, ω_{2}^{(K)}, …, ω_{m}^{(K)}, … are the frequencies of the eigenoscillations of problem (K) (with their multiplicity taken into account),

and U^{(K,1)}, U^{(K,2)}, …, U^{(K,m)}, … are the corresponding vector-functions, then

{ ω^{(K)}_{m}^{2} + κ_{0}^{2}}_{m=1}^{∞} and { V^{(K,m)}}_{m=1}^{∞} will be a spectrum of eigenvalues and a system of eigenvector-functions of Eq. (4.2) and vice versa. It can be as-

sumed that 0 ≤ ω_{1}^{(K)} ≤ ω_{2}^{(K)} ≤ …, ω_{1}^{(K)} > 0, lim_{m→∞} ω_{m}^{(K)} = ∞, and { V^{(K,m)}}_{m=1}^{∞}

is a system of orthonormalized vectors

∫_{Ω} V^{(K,m)}(y) · V^{(K,l)}(y) dy = δ_{ml}, m, l = 1, 2, 3 ….

Then { U^{(K,m)}}_{m=1}^{∞} will be a system of orthonormalized vector-functions with weight E, i.e.,

∫_{Ω} E U^{(K,m)}(y) · U^{(K,l)}(y) dy = δ_{ml}, m, l = 1, 2, 3 …, K=I,II,III,IV.

We have

Lemma 4.1. The matrix G^{(K)}(x, y, μ_{0} − κ_{0}^{2}) is the resolvent of the kernel

G^{(K)}(x, y, −κ_{0}^{2}) and the equality

G^{(K)}(x, y, −κ^{2}) − G^{(K)}(x, y, −κ_{0}^{2}) =

= μ_{0} ∫_{Ω} G^{(K)}(x, z, −κ_{0}^{2}) G^{(K)}(z, y, −κ_{0}^{2}) dz  (4.3)

is fulfilled, where κ > κ_{0} > 0, μ_{0} = κ_{0}^{2} − κ^{2} < 0.

Proof. Let

H^{(K)}(x, y, −κ^{2}) = G^{(K)}(x, y, −κ^{2})F − G^{(K)}(x, y, −κ_{0}^{2})F.

Then for x ≠ y the equalities

A(D_{x}, −κ_{0}^{2}) G^{(K)}(x, y, −κ^{2})F = (κ^{2} − κ_{0}^{2})E G^{(K)}(x, y, −κ^{2})F,

A(D_{x}, −κ_{0}^{2}) G^{(K)}(x, y, −κ_{0}^{2})F = 0

imply

A(D_{x}, −κ_{0}^{2}) H^{(K)}(x, y, −κ^{2}) = (κ^{2} − κ_{0}^{2})E G^{(K)}(x, y, −κ^{2})F.  (4.4)

Moreover, H^{(K)} satisfies the boundary condition

B^{(K)}(D_{z}, n(z)) H^{(K)}(z, y, −κ^{2}) = 0, z ∈ S, y ∈ Ω,

and the equality

H^{(K)}(x, y, −κ^{2}) =−∫_{Ω} G^{(K)}(x, z, −κ_{0}^{2}) · A(D_{x}, −κ_{0}^{2}) H^{(K)}(z, y, −κ^{2}) dz+

+ ∫_{S} { G^{(K)}(x, z, −κ_{0}^{2}) · P(D_{z}, n) H^{(K)}(z, y, −κ^{2}) −

− [P(D_{z}, n) G^{(K)}(x, z, −κ_{0}^{2})]^{T} · H^{(K)}(x, y, −κ^{2})}d_{z}S.  (4.5)

In (4.5) the integral over the surface S is equal to zero. Therefore by virtue of relations (4.4) and (4.5) we have

H^{(K)}(x, y, −κ^{2})=(κ_{0}^{2} − κ^{2}) ∫_{Ω}G^{(K)}(x, z, −κ_{0}^{2}) · E G^{(K)}(z, y, −κ^{2})F dz. (4.6)

By multiplying equality (4.6) from the left by the matrix F we obtain relation (4.3).

By Lemma 4.1 the matrix G^{(K)}(x, y, −κ^{2}) − G^{(K)}(x, y, −κ_{0}^{2}) can be decomposed into the series [14]

G^{(K)}(x, y, −κ^{2}) − G^{(K)}(x, y, −κ_{0}^{2}) =

=(κ_{0}^{2} − κ^{2}) ∑_{m=1}^{∞}(λ_{m}^{(K)}+κ^{2})^{−1}(λ_{m}^{(K)}+κ_{0}^{2})^{−1} V^{(K,m)}(x)× V^{(K,m)}(y), (4.7)

where λ_{m}^{(K)} = ω^{(K)}_{m}^{2}, K=I,II,III,IV, m = 1, 2, 3 …, x, y ∈ Ω, κ > κ_{0} > 0. (4.7) implies

lim_{x→y} Sp [ G^{(K)}(x, y, −κ^{2}) − G^{(K)}(x, y, −κ_{0}^{2})] =

= (κ_{0}^{2} − κ^{2}) ∑_{m=1}^{∞}(λ_{m}^{(K)} + κ^{2})^{−1}(λ_{m}^{(K)} + κ_{0}^{2})^{−1}|V (y)|^{2}.  (4.8)

Taking into account the equalities Sp G^{(K)} = Sp( G^{(K)}E) and (2.5), from (4.8) we shall obtain

\frac{1}{4π} (κ_{0} − κ)M + Sp \{[ g^{(K)}(y, y, −κ^{2}) − g^{(K)}(y, y, −κ_{0}^{2})]E\} =

= (κ_{0}^{2} − κ^{2}) ∑_{m=1}^{∞}(λ_{m}^{(K)} + κ^{2})^{−1}(λ_{m}^{(K)} + κ_{0}^{2})^{−1}| V^{(K,m)}(y)|^{2}.  (4.9)

By Lemma 3.5, from (4.9) we have

∑_{m=1}^{∞}(λ_{m}^{(K)} + κ^{2})^{−1}(λ_{m}^{(K)} + κ_{0}^{2})^{−1}| V^{(K,m)}(y)|^{2} ∼ \frac{M}{4πκ} , κ → ∞. (4.10)

By the technique (a Tauber type theorem) used in [17, 19, 21], we obtain

∑_{ω_{m}^{(K)}≤t} | V^{(K,m)}(y)|^{2} ∼ \frac{M}{6π^{2}} t^{3}, t → ∞.  (4.11)

By (4.11) we can write the following formula for asymptotic behavior of eigen vector-functions of problem (K):

∑_{ω_{m}^{(K)}≤t} |F U^{(K,m)}(y)|^{2} ∼ \frac{M}{6π^{2}} t^{3}, t → ∞, K=I,II,III,IV.  (4.12)

Integrating equality (4.9) in the domain Ω and applying the fact that the

vectors of the system { V^{(K,m)}}_{m=1}^{∞} are orthonormal, also recalling the results of Lemma 3.5, we obtain

∑_{m=1}^{∞} [(λ_{m}^{(K)} + κ^{2})(λ_{m}^{(K)} + κ_{0}^{2})]^{−1} = \frac{1}{4π} |Ω|Mκ^{−1} + O(κ^{−2−2δ}), (4.13)

0 < δ < \frac{1}{2},

where |Ω| is the volume of Ω.

Similarly, like in the classical theory of elasticity [17, 19] and the couplestress theory of elasticity [21], from (4.13) we obtain the formula for asymptotic distribution of eigenoscillation frequencies of problem (K):

N^{(K)}(t) ∼ \frac{1}{6π^{2}} |Ω|Mt^{3}, t → ∞,  (4.14)

where N^{(K)}(t) = ∑_{ω_{m}^{(K)}≤t} 1 is the number of eigenoscillation frequencies ω_{m}^{(K)}

not greater than t (K=I,II,III,IV).

Thus we have proved

Theorem 4.1. For all the internal boundary value problems of the shift model of the linear theory of a mixture of two isotropic materials, the asymptotic behavior of eigenoscillation and eigenvector-function frequencies are expressed by formulas (4.14) and (4.12), respectively.

Remark 4.1. By virtue of (4.14) the asymptotic distribution of eigenoscillation frequencies does not depend on the form of an elastic mixture, but depends on its volume. With Remark 1.2 taken into account, the number M defined by (2.6) is the sum of the inverse cubes of velocities of plane waves, propagating in a mixture of two isotropic elastic materials, for α = 0.

Remark 4.2. The Weyl formula for the asymptotic distribution of eigenoscillation frequencies in the classical theory of elasticity for a three-dimen-sional isotropic body is written as [16]

N(t) ∼ \frac{1}{6π^{2}} |Ω|M_{0}t^{3}, t → ∞,  (4.15)

where N(t) is the number of eigenoscillation frequencies not greater than t, M_{0} = ρ^{3/2}[(λ+2μ)^{−3/2} +2μ^{−3/2}] = v_{1}^{−3} +2v_{2}^{−3}. By Remark 1.2 the number M_{0} is the sum of the inverse cubes of velocities of plane waves propagating in an isotropic body. Formula (4.14) is a corollary of formula (4.14). Indeed, if we consider an isotropic body as a mixture of two isotropic materials of the same kind, then the elastic constants and densities will satisfy the conditions

\begin{matrix}a_{1} = a_{2} = μ, b_{1} = b_{2} = λ + μ, c = d = 0, \\ ρ_{11} = ρ_{22} = ρ, ρ_{12} = 0. \end{matrix}  (4.16)

By virtue of (1.9), (1.10), (4.16) we have c_{1} = c_{2} = v_{1}, c_{3} = c_{4} = v_{2}, M = 2M_{0}. With conditions (4.16) taken into account, system (1.1) takes the form

μΔu′ + (λ + μ) grad div u′ + ρω^{2}u′ = 0,  (4.17)
μΔu′′ + (λ + μ) grad div u′′ + ρω^{2}u′′ = 0,  (4.18)

and the boundary condition in problem (I) becomes

u′(z) = 0,  (4.19)
u′′(z) = 0, z ∈ S.  (4.20)

Obviously, problems (4.17)–(4.20) and (4.17), (4.19) have the same eigenoscillation frequencies, but the multiplicity of these frequencies will be two times higher for problem (4.17)–(4.20) than for problem (4.17), (4.19). Thus we have

N(t) = \frac{1}{2} N^{(I)}(t) ∼ \frac{1}{2} \frac{1}{6π^{2}} |Ω|Mt^{3} = \frac{1}{6π^{2}} |Ω|M_{0}t^{3}, t → ∞.

References

1. D. G. Natroshvili, A. J. Jagmaidze, and M. Zh. Svanadze, Some problems of the theory of elastic mixtures. (Russian) Tbilisi Univ. Press, Tbilisi, 1986.

2. L. P. Khoroshun and N. S. Soltanov, Thermoelasticity of two-compo-nent mixtures. (Russian) Naukova Dumka, Kiev, 1984.

3. Ya. Ya. Rushchitskii, Elements of mixture theory. (Russian) Naukova Dumka, Kiev, 1991.

4. I. G. Filippov, Dynamical theory of a relative flow of multicomponent media. (Russian) Prikl. Mekhanika 7(1971), No. 10, 92–99.

5. B. Lempriere, On practicability of analyzing waves in composites by the theory of mixtures. Lockheed Palo Alto Research Laboratory. Report No. LMSC-6-78-69-21 (1969), 76–90.

6. H. D. McNiven and Y. A. Mengi, A mathematical model for the linear dynamic behavior of two-phase periodic materials. Int. J. Solids and Struct. 15(1979), No. 4, 571–580.

7. T. R. Steel, Applications of a theory of interacting continua. Quart. J. Mech. and Appl. Math. 20(1967), No. 1, 57–72.

8. M. Zh. Svanadze, Representation of a general solution of the equation of steady-state oscillations of two-component elastic mixtures. (Russian) Prikl. Mekhanika 29(1993), No. 12, 22–29.

9. M. Zh. Svanadze, The uniqueness of solutions of stable oscillation problems of the linear theory of a two-component elastic mixture. (Russian) Bull. Acad. Sci. Georgia 145(1992), No. 1, 51–54.

10. M. Zh. Svanadze, Investigation of boundary value problems of steady-state oscillations of the theory of elastic mixtures. 10th Conference of Problems and Methods in Math. Physics, September 13–17, 1993, Chemnitz, Germany, Abstracts, p. 60.

11. M. Zh. Svanadze, Uniqueness theorem for solutions of the internal boundary value problems of steady-state oscillations of the linear theory of elastic mixtures. Proc. I. Vekua Inst. Appl. Math. Tbilisi State Univ. 46(1992), 179–190.

12. H. Weyl, Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung. J. Reine und Angew. Math. 141(1912), 1–11.

13. H. Weyl, Über die Randwertaufgabe der Strahlungstheorie und asymptotische Spektralgesetze. J. Reine und Angew. Math. 143(1913), 177–202.

14. R. Courant and D. Hilbert, Methoden der mathematischen Physik. Julius Springer, Berlin, B.1, 2, 1931, 1937.

15. T. Carleman, Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes. Comptes Rendus des Mathématiciens Scandinaves á Stockholm 14–18 Aout 1934, 34–44, Lund, 1935.

16. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers. Rendiconti del Circolo Matematico di Palermo 39(1915), 1–49.

17. A. Plejel, Propriétés asymptotiques des fonctions et valeurs propres de certains problems de vibrations. Arkiv för Math., Astr. och Fysik. 27A(1940), 1–100.

18. H. Niemeyer, Über die elastischen Eigenschwingungen endlicher Körper. Arch. Rat. Mech. Ann. 19(1965), 24–61.

19. T. V. Burchuladze, To the theory of boundary value problems of oscillation for an elastic body. (Russian) Proc. Tbilisi University, Mat. Mech. Astron. 64(1957), 215–240.

20. T. V. Burchuladze, On the asymptotic behavior of eigenfunctions of some boundary value problems of an anisotropic elastic body. (Russian) Bull. Acad. Sci. Georgian SSR 23(1959), No. 3, 265–272.

21. R. G. Dikhamindzhia, Asymptotic distribution of eigenfunctions and eigenvalues of some basic problems of vibration of the moment theory of elasticity. (Russian) Trudy Tbiliss. Matem. Inst. Razmadze 73(1983), 64–71.

22. C. Müller and H. Niemeyer, Greensche Tensoren und asymptotische Gesetze der elekromagnetischen Hohlraumschwingungen. Arch. Rat. Mech. Ann. 7(1961), 305–348.

23. H. Niemeyer, Eine Verschärfung der asymptotischen Gesetze elektomagnetischen Hohlraumschwingungen. Arch. Rat. Mech. Ann. 7(1961), 412–433.

24. W. Gromes, Über das asymptotische Verhalten der Spektralfunktion elliptischer Systeme. Math. Zeitschrift 118(1970), No. 4, 254–270.

25. R. Brübach, Über die Spektralmatrix elliptischer Systeme. Math. Zeitschrift 140(1974), No. 3, 231–244.

26. 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.

27. M. Zh. Svanadze, Fundamental solutions of equations of stable oscillation and pseodo-oscillation of a two-component elastic mixture. (Russian) Proc. I. Vekua Inst. Appl. Math. Tbilisi State Univ. 39(1990), 227–240.

28. I. G. Petrashen, Elastic wave propagation in anisotropic media. (Russian) Nauka, Leningrad, 1980.

(Received 06.10.1994)

Author’s address: I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia