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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
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.
177
c 1996 Plenum Publishing Corporation
1072-947X/96/0300-0177$09.50/0
178
M. SVANADZE
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.
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-functions 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 M¨
uller and Niemeyer in [22,
23].
Using V. Avakumovi´c’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-functions.
1. Formulation of Boundary Value Problems
Let the finite domain Ω of the three-dimensional Euclidean space R3
be bounded by the surface S, S ∈ L2 (ε), 0 < ε ≤ 1 [26], 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 = (f1 , f2 , . . . , fk ) and
Pk
ϕ = (ϕ1 , ϕ2 , . . . , ϕk ) will be denoted by f · ϕ =
j=1 fj ϕj , where ϕj is
the complex-conjugate to the number ϕj , f × ϕ = kψml kk×k , ψml = fm ϕl ,
Pk
m, l = 1, k, |f | = ( l=1 fl2 )1/2 . The product of the matrix A = kAml kp×k
Pk
Pk
Pk
by f denotes the vector Af = ( j=1 A1j fj , k=1 A2j fj , . . . , j=1 Apj fj );
The trace of the square matrix
AT is the transposition of the matrix A. P
k
A = kAml kk×k will be denoted by Sp A = j=1 Ajj .
In the absence of mass force, the system of stationary oscillation equations in the shift model of the linear theory of two-component elastic mix-
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
179
tures has the form [3, 5, 6, 8, 27]
a1 ∆u′ + b1 grad div u′ + c∆u′′ + d grad div u′′ −
−α(u′ − u′′ ) + ω 2 ρ11 u′ − ω 2 ρ12 u′′ = 0,
c∆u′ + d grad div u′ + a2 ∆u′′ + b2 grad div u′′ +
(1.1)
+α(u′ − u′′ ) − ω 2 ρ12 u′ + ω 2 ρ22 u′′ = 0,
where u′ = (u′1 , u′2 , u′3 ), u′′ = (u′′1 , u′′2 , u3′′ ) are partial displacements, ∆ is the
three-dimensional Laplace operator, ω is an oscillation frequency, ω > 0,
α ≥ 0,
(−1)j ρ3−j α2
,
ρ1 + ρ2
ρjj = ρj + ρ12 , j = 1, 2, c = µ3 + λ5 ,
ρ2 α2
ρ1 α2
= µ 3 + λ4 − λ5 +
, α2 = λ 3 − λ 4 ,
d = µ3 + λ3 + λ5 −
ρ1 + ρ 2
ρ1 + ρ2
µ1 , µ2 , µ3 , λ1 , λ2 , . . . , λ5 are elastic constants of the mixture [1, 7].
aj = µj − λ5 , bj = µj + λj + λ5 +
In the sequel it will be assumed that the following conditions are fulfilled
[1, 7]:
ρ2 α2
2
+ µ1 > 0, λ5 ≤ 0,
ρ1 + ρ2
3
ρ11 > 0, ρ11 ρ22 > ρ212 ,
ρ1 α2
ρ2 α2
2
2
+ µ1 λ2 +
+ µ2 >
λ1 −
ρ1 + ρ2
3
ρ1 + ρ2
3
2 2
ρ 1 α2
+ µ3 .
> λ3 −
ρ1 + ρ2
3
µ1 > 0, µ1 µ2 > µ23 , λ1 −
System (1.1) can be rewritten in the matrix form as
A(Dx , ω 2 )U (x) ≡ A(Dx ) + αE0 + ω 2 E U (x) = 0,
(1.2)
(1.3)
where
U = (U1 , U2 , . . . , U6 ) = (u′ , u′′ ), x = (x1 , x2 , x3 ) ∈ Ω,
∂
∂
∂
,
,
Dx =
,
∂x1 ∂x2 ∂x3
(1)
A (Dx ) A(2) (Dx )
(j)
A(Dx ) = (2)
, A(j) (Dx ) = kAkl (Dx )k3×3 ,
A (Dx ) A(3) (Dx )6×6
(1)
Akl (Dx ) = a1 ∆δkl + b1
∂2
∂2
(2)
, Akl (Dx ) = c∆δkl + d
,
∂xk ∂xl
∂xk ∂xl
180
M. SVANADZE
∂2
(3)
, j, k, l = 1, 2, 3,
Akl (Dx ) = a2 ∆δkl + b2
∂xk ∂xl
−I I
ρ11 I −ρ12 I
=
E0 =
,
E
, I = kδkl k3×3 ,
I −I
−ρ12 I ρ22 I
6×6
6×6
δkl is the Kronecker symbol.
A vector-function U is called regular in the domain Ω if Uj ∈ C 2 (Ω) ∩
1
C (Ω) (j = 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
(K)
(K)
B (Dz , n(z))U (z) ≡
where
lim
Ω∋x→z∈S
B (Dx , n(z))U (x) = 0,
I
P(Dx , n(z))
(K)
P(Dx , n(z)) + σI
B (Dx , n(z)) =
I
I
(1)
(2)
(3)
(2)
P + P
P +P
for K=I,
for K=II,
for K=III,
,
for K=IV,
6×6
I = kδkl k6×6 , σ > 0, P(Dx , n(z)) is the stress operator in the theory of
elastic mixtures [1],
P (1) P (2)
(j)
P(Dx , n(z) =
, P (j) = kPkl k3×3 ,
P (2) P (3)
6×6
∂
∂
ρ2 α2
∂
(1)
Pkl (Dx , n(z)) = (µ1 −λ5 )δkl
+(µ1 +λ5 )nl
+ λ1 −
,
nk
∂n
∂xk
ρ1 +ρ2
∂xl
∂
ρ1 α2
∂
∂
(2)
+(µ3 −λ5 )nl
+ λ3 −
,
nk
Pkl (Dx , n(z)) = (µ3 +λ5 )δkl
∂n
∂xk
ρ1 +ρ2
∂xl
∂
ρ2 α2
∂
∂
(3)
+(µ2 +λ5 )nl
+ λ2 −
,
Pkl (Dx , n(z)) = (µ2 −λ5 )δkl
nk
∂n
∂xk
ρ1 +ρ2
∂xl
n = (n1 , n2 , n3 ) 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(Dx , −κ 2 )U (x) = 0, x ∈ Ω,
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
181
and the boundary condition
(K)
B (Dz , 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,δ1 , 0 < δ1 ≤ 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]
Z
Z
U (x) · A(Dx )U (x) + W (U, U ) dx = U (z) · P(Dz , n(z))U (z) dz S, (1.4)
S
Ω
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
a = a1 + b1 , b = a2 + b2 , c0 = c + d, d1 = ab − c20 ,
d2 = a1 a2 − c2 , ρ0 = ρ11 ρ22 − ρ212 , ρ3 = ρ11 + ρ22 − 2ρ12 ,
q1 = a + b + 2c0 , q2 = aρ22 + bρ11 + 2c0 ρ12 ,
(1.5)
q3 = a1 + a2 + 2c, q4 = a1 ρ22 + a2 ρ11 + 2cρ12 .
On account of (1.2) we obtain by (1.5)
aj > 0, a > 0, b > 0, dj > 0,
ρ0 > 0, ρ3 > 0, ql > 0, j = 1, 2, l = 1, 4.
Remark 1.1. If α = 0 then (1.1) implies
div u′
2
2
∆ + k 1 ∆ + k2
= 0,
div u′′
rot u′
= 0,
∆ + k32 ∆ + k42
rot u′′
(1.6)
(1.7)
where k12 , k22 and k32 , k42 are the roots of the square equations d1 ξ 2 − ω 2 q2 ξ +
ω 4 ρ0 = 0 and d2 ξ 2 − ω 2 q4 ξ + ω 4 ρ0 = 0, respectively. By virtue of conditions
(1.6) we have kj2 > 0 (j = 1, 4). Assume that kj > 0 (j = 1, 4). By (1.7)
182
M. SVANADZE
it is clear that k1 , k2 and k3 , k4 are the wave numbers of longitudinal and
transverse waves, respectively [3]. Let
cj = ωkj−1 , j = 1, 4.
(1.8)
The constants c1 and c2 will be velocities of longitudinal waves, while c3
and c4 the velocities of transverse waves [3]. Clearly, c21 , c22 and c32 , c24 are
the roots of the equations
ρ0 ξ 2 − q2 ξ + d1 = 0,
2
ρ0 ξ − q4 ξ + d2 = 0.
(1.9)
(1.10)
Remark 1.2. As is well known [28], by the classical theory of elasticity,
in an p
isotropic body one longitudinal wave propagates with the velocity
v1 = p (λ + 2µ)ρ−1 , and two transverse waves with the equal velocities
v2 = µρ−1 (here λ, µ are the Lam´e 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 c1 and c2 , and four transverse waves with
the velocities c3 and c4 (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(Dx , −κ 2 )U (x) = 0
is constructed in [27] in terms of elementary functions. It has the form
(1)
Γ (x, −κ 2 ) Γ(2) (x, −κ 2 )
2
Γ(x, −κ ) = (2)
,
Γ (x, −κ 2 ) Γ(3) (x, −κ 2 )6×6
(2.1)
(j)
Γ(j) = kΓkl k3×3 , j = 1, 2, 3,
n1
(1)
(a2 ∆ − β3 )(∆ − κ12 )(∆ − κ22 )δkl +
Γkl (x, −κ 2 ) =
d2
(1)
(1)
(1)
∂2 o
+ r1 ∆2 + r2 ∆ + r3
γ,
∂xk ∂xl
n 1
(2)
Γkl (x, −κ 2 ) = − (c∆+β2 )(∆−κ12 )(∆−κ22 )δkl +
d2
(2)
(2)
(2)
∂2 o
+ r1 ∆2 + r2 ∆ + r3
γ,
∂xk ∂xl
n1
(3)
(a1 ∆ − β1 )(∆ − κ12 )(∆ − κ22 )δkl +
Γkl (x, −κ 2 ) =
d2
(3)
(3)
(3)
∂2 o
+ r1 ∆2 + r2 ∆ + r3
γ,
∂xk ∂xl
k, l = 1, 2, 3,
(2.2)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
183
where
β1 = α + κ 2 ρ11 , β2 = α + κ 2 ρ12 , β3 = α + κ 2 ρ22 ,
(3)
a2 (2)
c
a
a1
c0
b
− , r1 = − + , r1 =
− ,
d1
d2
d1
d2
d1
d2
(1)
1
r2 =
2β2 (a2 d − cb2 ) + β3 (2a2 b1 − 2cd − b1 b2 − d2 ) ,
d1 d 2
(2)
1
r2 =
β1 (a1 d − b2 c) + β2 (a2 b1 − 2cd + ab2 − d2 ) + β3 (a1 d − b1 c) ,
d1 d 2
(3)
1
r2 =
β1 (2a1 b2 − 2cd + b1 b2 − d2 ) + 2β2 (a1 d − b1 c) ,
d1 d 2
(1)
1 2
r3 = −
b1 β3 + 2dβ2 β3 + b2 β22 ,
d1 d2
(2)
1
b1 β2 β3 + b2 β1 β2 + dβ1 β3 + dβ22 ,
r3 = −
d1 d2
4
X
(3)
1 2
2
r3 = −
ηj γj ,
bβ2 + 2dβ1 β2 + b2 β1 , γ =
d1 d2
j=1
(1)
r1
=
ηj =
4
Y
l=1
l6=j
(κj2 − κl2 )−1 , γj (x, −κ 2 ) = −
e−κj |x|
, j = 1, 4,
4π|x|
κ12 , κ22 and κ32 , κ42 are, respectively, the roots of the following square equations
d1 ξ 2 − (aβ3 + bβ1 + 2c0 β2 )ξ + β1 β3 − β22 = 0
and
d2 ξ 2 − (a1 β3 + a2 β1 + 2cβ2 )ξ + β1 β3 − β22 = 0.
By virtue of (1.6) we have κj2 > 0 (j = 1, 4). Assume that κj > 0 (j = 1, 4)
and introduce the notation
−1
−1 (0)
(0)
2
2
) ,
) , ql = d2 κl2 (κl2 −κ7−l
qj = d1 κj2 (κj2 −κ3−j
(1)
qj
(1)
ql
(0)
(2)
(0)
(3)
(0)
= (bκj2 −β3 ) qj , qj = −(c0 κj2 +β2 ) qj , qj = (aκj2 −β1 ) qj ,
(0)
(2)
(0)
(3)
(0)
= (a2 κj2 −β3 ) ql , ql = −(cκl2 +β2 ) ql , ql = (a1 κl2 −β1 ) ql ,
(1)
(2)
(3)
pk = ρ11 qk −2ρ12 qk +ρ22 qk , j = 1, 2, l = 3, 4, k = 1, 4.
We have
Lemma 2.1. The matrix Γ(x, −κ 2 ) has the following properties:
(a) ΓT (x, −κ 2 ) = Γ(x, −κ 2 );
(b) Γ(−x, −κ 2 ) = Γ(x, −κ 2 );
(2.3)
184
M. SVANADZE
(c) A(Dx , −κ 2 )Γ(x, −κ 2 ) = δ(x)I, where δ(x) is the Dirac distribution
function;
(d) for x 6= 0 the matrices Γ(1) , Γ(2) , Γ(3) have the form
Γ(k) (x, −κ 2 ) = grad div
− rot rot
2
X
(k)
qj γj (x, −κ 2 ) −
j=1
4
X
l=3
(k)
ql γl (x, −κ 2 ),
k = 1, 2, 3;
(2.4)
√
1
Γ(x, −κ 2 ) − Γ(x, −κ02 ) E =
(2.5)
κM + O( κ),
x→0
4π
κ > κ0 > 0, κ → ∞;
s
∂
c∗ e−α0 κ|x|
(f) s1 s2 s3 Γkl (x, −κ 2 ) <
,
∂x1 ∂x2 ∂x3
|x|1+s
where c∗ = const > 0, s = s1 + s2 + s3 , s1 , s2 , s3 are nonnegative integer
numbers, α0 is a positive number not depending on κ and x,
(e)
lim Sp
−3
−3
−3
M = c−3
1 + c2 + 2(c3 + c4 ).
(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
∆γj (x, −κ 2 ) = κj2 γj (x, −κ 2 ),
1
Iγj (x, −κ 2 ) = 2 grad div − rot rot γj (x, −κ 2 ), x 6= 0, j = 1, 4,
κj
we find by (2.2) that
nh 1
2
2
2
2
2
2 (a2 κj − β3 )(κj − κ1 )(κj − κ2 ) +
d
κ
2
j
j=1
i
(1)
(1)
(1)
+ r1 κj4 + r2 κj2 + r3 grad div −
o
(a2 κj2 − β3 )(κj2 − κ12 )(κj2 − κ22 ) rot rot γj (x, −κ 2 ). (2.7)
Γ(1) (x, −κ 2 ) =
−
1
d2 κj2
4
X
ηj
Using the relations
(1)
(1)
(1)
1
(a2 κj2 − β3 )(κj2 − κ12 )(κj2 − κ22 ) + r1 κj4 + r2 κj2 + r3 =
d2 κj2
1
=
(bκj2 − β3 )(κj2 − κ32 )(κj2 − κ42 ), j = 1, 4,
d1 κj2
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
185
which are easy to verify, from (2.7) we obtain
h 1
2
2
2
2
2
2 (bκj − β3 )(κj − κ3 )(κj − κ4 ) grad div −
d
κ
1
j
j=1
i
(a2 κj2 − β3 )(κj2 − κ12 )(κj2 − κ22 ) rot rot γj (x, −κ 2 ) =
Γ(1) (x, −κ 2 ) =
4
X
ηj
−
1
d2 κj2
=
2
X
j=1
1
(bκj2 − β3 )(κj2 − κ32 )(κj2 − κ42 )ηj grad div γj (x, −κ 2 ) −
d1 κj2
4
X
1
(a2 κj2 − β3 )(κl2 − κ12 )(κl2 − κ22 )ηl rot rot γl (x, −κ 2 ) =
d2 κj2
−
l=3
= grad div
2
4
X
X
(1)
(1)
ql γl (x, −κ 2 ).
qj γj (x, −κ 2 ) − rot rot
j=1
l=3
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
(1)
Φ
Φ(2)
(l)
Φ = (3)
, Φ(l) = kΦkj k3×3 , l = 1, 4,
(4)
Φ
Φ
6×6
Φ(1) = ρ11 Γ(1) − ρ12 Γ(2) ,
Φ(3) = ρ11 Γ(2) − ρ12 Γ(3) ,
Φ(2) = −ρ12 Γ(1) + ρ22 Γ(2) ,
Φ(4) = −ρ12 Γ(2) + ρ22 Γ(3) .
Clearly, Sp Φ = Sp(ρ11 Γ(1) − 2ρ12 Γ(2) + ρ22 Γ(3) ). Hence on account of (2.3),
(2.4) we obtain
4
2
i
h
X
X
Sp Φ(x, −κ 2 ) = Sp grad div
pl γl (x, −κ 2 ) =
pj γj (x, −κ 2 ) − rot rot
j=1
=∆
2
hX
j=1
=
2
X
j=1
Using the equalities
have
l=3
pj γj (x, −κ 2 ) + 2
pj κj2 γj (x, −κ 2 ) + 2
P2
j=1
l=3
4
X
l=3
pj κj2 = q2 d−1
1 ,
i
pl γl (x, −κ 2 ) =
pl κl2 γl (x, −κ 2 ).
P4
l=3
(2.8)
pl κl2 = q4 d−1
2 , from (2.8) we
i
X
1 hX
pj κj2 + 2
pl κl2 − p5 (κ)|x| +
4π|x| j=1
2
Sp Φ(x, −κ 2 ) = −
4
X
4
l=3
186
M. SVANADZE
i
2q4
1 h q2
+
− p5 (κ)|x| + O(|x|), |x| ≪ 1, (2.9)
4π|x| d1
d2
P2
P4
where p5 (κ) = j=1 pj κj3 + 2 l=3 pl κl3 . The relation (2.9) readily implies
+O(|x|) = −
1
p5 (κ) − p5 (κ0 ) , κ0 > 0. (2.10)
lim Sp Φ(x, −κ 2 ) − Φ(x, −κ02 ) =
x→0
4π
Let us now calculate the difference p5 (κ) − p5 (κ0 ) for κ > κ0 > 0,
κ → ∞. Assume that τ12 , τ22 and τ32 , τ42 are the roots of the square equations
d1 ξ 2 − κ 2 q2 ξ + κ 4 ρ0 = 0 and dξ 2 − κ 2 q4 ξ + κ 4 ρ0 = 0, respectively. By (1.6)
we have τj2 > 0 (j = 1, 4). Assuming that τj > 0 (j = 1, 4), we obtain
(j = 1, 4), where cj (j = 1, 4) are defined by (1.8). Taking into
τj = κc−1
j
account the relations
τ12 + τ22 = κ 2 q2 d1−1 , τ12 τ22 = κ 4 ρ0 d−1
1 ,
κ12 + κ1 κ2 + κ22 = τ12 + τ1 τ2 + τ22 + O(κ),
i
τ1 + τ2 = (κ1 + κ2 ) 1 + O(κ −1/2 ) , κ → ∞,
we find by (2.3) that
2
X
pj κj3 =
j=1
hq
αρ3 i
1
2κ 2 ρ0
2
(κ12 + κ1 κ2 + κ22 ) −
−
=
κ1 + κ2 d1
d1
d1
αρ3
1 τ1 + τ2
(τ 3 + τ23 ) +
=
κ 2 κ1 + κ2 1
d1 (κ1 + κ2 )
√
= κ(c1−3 + c2−3 ) + O( κ), κ → ∞.
=
Similarly, one can show that
Therefore the equality
P4
l=3
√
−3
pl κl2 = κ(c−3
3 +c4 )+O( x), x → ∞.
√
p5 (κ) − p5 (κ0 ) = κM + O( κ), κ > κ0 > 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
(K)
(K)
G (x, y, −κ 2 ) = −Γ(x − y, −κ 2 ) + g (x, y, −κ 2 )
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
187
will be called the Green tensor of problem (K) (K=I, II, III, IV) if the
(K)
matrix g satisfies the homogeneous equation
(K)
A(Dx , −κ 2 ) g (z, y, −κ 2 ) = 0, x, y ∈ Ω,
and the boundary condition
(K)
(K)
(K)
B (Dz , n(z)) g (z, y, −κ 2 ) = B (Dz , n(z))Γ(z − y, −κ 2 ), z ∈ S, y ∈ Ω.
(K)
The matrix g (K=I, II, III, IV) is a solution of problem (K)∗f with
(K)
(K)
a special boundary value f = B Γ. The existence and uniqueness of g
follows from Lemma 1.1.
e
e = (e
u′ , u
e′′ ). Assume that U, U
e′′2 , u
e′′3 ), U
e′′ = (e
Let u
u′′1 , u
u′1 , u
e′2 , u
e′ = (e
e′3 ), u
are the vector-functions with real components. Introduce the notation [1, 7]
ε′lj =
εe′lj =
hlj =
e
hlj =
e) =
W (U, U
+
+
1 ∂u′j
1 ∂uj′′
∂u′l ′′
∂u′′l
+
+
, εlj =
,
2 ∂xl
∂xj
2 ∂xl
∂xj
u′j
u′′j
1 ∂e
∂e
ul′ ′
∂e
u′′l
1 ∂e
+
+
, εelj =
,
2 ∂xl
∂xj
2 ∂xl
∂xj
∂uj′′
∂u′l
∂u′′l
1 ∂u′j
−
+
−
,
2 ∂xl
∂xj
∂xl
∂xj
u′j
u′′j
∂e
∂e
ul′
∂e
u′′l
1 ∂e
, l, j = 1, 2, 3,
−
+
−
2 ∂xl
∂xj
∂xl
∂xj
3 h
X
′
′′ ′
(λ1 εll
+ λ3 ε′′ll )e
ε′jj + 2µ1 ε′lj εe′lj + 2µ3 εlj
εelj +
l,j=1
(λ4 ε′ll
′′
′′ ′′
+
εjj + 2µ3 ε′lj εe′′lj + 2µ2 ε′′lj εelj
)e
+ λ2 εll
i
α2
′′
− εe′jj ) − 2λ5 hlj e
hlj .
εjj
(ρ2 ε′ll + ρ1 ε′′ll )(e
ρ1 + ρ2
Consider the functions
Z
e dx,
e]=
e + κ2 U E U
e )−αU E0 U
L[U, U
W (U, U
Ω
R
e +κ 2 U E U
e ]dx ≡ L[U ] for
[W
U
(U,
)−αU E0 U
Ω
R
L[U ]−2 S U (z) · PΓj (z − y, −κ 2 )dz S
for
(K)
L [U ] = (II)
R
L [U ]+σ [U (z)−Γ (z − y, −κ 2 )]2 d S
for
j
z
R S
e j (z − y, −κ 2 )dz S
L[U ]−2 S u′′ (z) · PΓ
for
(3.1)
K=I,
K=II,
K=III,
K=IV,
(3.2)
188
M. SVANADZE
where Γj is the jth column of the matrix Γ, j is some fixed number, j = 1, 6,
e = kP (1) + P (2) , P (2) + P (3) k6×6 , σ > 0.
P
It is assumed that
(K)
R = U : U ∈ C 2 (Ω) ∩ C 1 (Ω), B U (z) =
(K)
(K)
= B Γj (z − y, −κ 2 ), z ∈ S, y ∈ Ω
(3.3)
(K)
is the set of admissible vector-functions for the functional L (K=I,II,III,IV).
Let [17]
n
h
r m il o
e − y, −κ 2 ) = 1 − 1 −
Γ(x
Γ(x − y, −κ 2 ),
(3.4)
ρy (x)
where r = |x − y|, ρy (x) = max{r, ℓy }, ℓy is the distance from the point y
to S, while m, l are natural numbers.
The following lemmas hold.
e has the properties:
Lemma 3.1. The matrix Γ
(K)
e j ∈ R , j = 1, 6, K=I,II,III,IV;
a) Γ
e j (x − y, −κ 2 ) = Γj (x − y, −κ 2 ) for r ≥ ℓy ;
b) Γ
∂s
e kj (x − y, −κ 2 ) < const e−α0 κr rm−s−1 for r < ℓy ,
c) s1 s2 s3 Γ
∂x1 ∂x2 ∂x3
ℓm
y
m ≥ s + 1, l ≥ s + 1, s = s1 + s2 + s3 , s1 , s2 , s3 are negative numbers, α0
e j is the jth column of
is a positive number not depending on κ, x, y, and Γ
e
the matrix Γ, j, k = 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).
e are vector-functions, regular in Ω, then the
Lemma 3.2. If U and U
functionals L and L have the properties:
(a) L[U ] ≥ 0. If L[U ] = 0, then U (x) ≡ 0, x ∈ Ω;
e ] = L[U
e , U ];
(b) L[U, U
(c) L[U, U ] = L[U ];
e
e
e
(d) L[U
Z + U ] = L[U ] + 2L[U, U ] + L[U ]; Z
e · PU dS;
e · A(Dx , −κ 2 )U dx + L[U, U
e] =
U
(e)
(3.5)
U
S
Ω
Z
e · P (K)
e , (K)
U
gj dS, K=I,II,III,IV;
(3.6)
gj ] =
(f) L[U
S
(I)
e , (I)
e = U − (I)
gj ] = 0 for U
(g) L[U
gj , U ∈ R, j = 1, 6.
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
189
Proof. The validity of properties (a), (b), (c), (d) is obvious by (3.1). Relation (3.5) follows from the Green formula [1]
Z
Z
e · PU dS.
e
e
U
U · A(Dx )U + W (U, U ) dx =
Ω
S
(I)
(K)
If U = gj , then from (3.5) we obtain equality (3.6). If U ∈ R, then
e (z) = 0, z ∈ S.
e = U − (I)
the vector U
gj satisfies the boundary condition U
e , (I)
gj ] = 0.
Therefore (3.6) implies L[U
(K)
Lemma 3.3. The functional L has a minimum value only on the vector
(K)
gj ,
i.e.,
(K)
(K) (K)
min L [U ] = L [ gj ], K =I,II,III,IV.
(3.7)
(K)
U∈ R
(I)
e ≡ U − (I)
gj , then by Lemma 3.2 we have
Proof. If U ∈ R, U
(I)
(I)
(I)
e ] = L[gj ] + 2L[ g j , U
e ] + L[U
e] =
L[U ] = [L[gj + U
(III)
(I)
e ] ≥ L[(I)
= L[gj ] + L[U
gj ].
(III)
e = U − gj . Then
Let U ∈ R , gj = gj , U
Z
Z
(III)
e
e
e · PΓj dS +
L [U ] = L[gj ] + 2L[gj , U ]+L[U ]−2 gj PΓj dS −2 U
S
S
Z
Z
Z
e 2 dS.
e dS +σ
+σ [gj −Γj ]2 dS + 2σ (gj −Γj ) · U
(3.8)
U
S
S
S
By virtue of (3.2), (3.6) and the boundary condition Pgj + σgj = PΓj +
σΓj we find from (3.8) that
Z
(III)
(III)
(III)
e] + σ
e 2 dS ≥ L [gj ].
L [U ] ≡ L [gj ] + L[U
U
S
Quite similarly, equality (3.7) is proved for K=II,IV. Property (a) of
(K)
Lemma 3.2 implies that the functional in the set R has a minimum only
(K)
on gj (K=I,II,III,IV).
(I) (II) (III) (IV)
Lemma 3.4. The vector-functions gj , gj , gj , gj satisfy the following
relations:
190
M. SVANADZE
(II) (III)
L [ gj ]
(a)
(b)
(III) (III)
≤ L [ gj ],
(3.9)
(I)
(I)
gjj (y, y, −κ 2 ) = −L[gj ]+
Z
+ Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S;
(3.10)
S
(c)
(K) (K)
(K)
gjj (y, y, −κ 2 ) = − L [ gj ]−
Z
− Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S, K=II,III;
(3.11)
S
(IV) (IV)
(IV)
gjj (y, y, −κ 2 ) = − L [ gj ]+
(d)
(e)
Z
[Γ′j Pj′ −2Γ′′j Pj′ −Γ′′j Pj′′ ] dz S;
(3.12)
S
(III) (I)
(I)
gjj (y, y, −κ 2 ) = − L [gj ]−
Z
− Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S;
(3.13)
S
(I)
(III)
(II)
gjj (y, y, −κ 2 ) ≤ gjj (y, y, −κ 2 ) ≤ gjj (y, y, −κ 2 ),
(f)
(IV)
(I)
gjj (y, y, −κ 2 ) ≤ gjj (y, y, −κ 2 ),
(3.14)
(3.15)
where y ∈ Ω, Γ′j = (Γ1j , Γ2j , Γ3j ), Γ′′j = (Γ4j , Γ5j , Γ6j ), Pj′ = (P (1) +
P (2) )Γj′ , Pj′′ = (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
Z (K)
(K)
T
G (y, z, −κ 2 )PU (z) − P G (z, y, −κ 2 ) U (z) dz S −
U (y) =
−
Z
S
(K)
Ω
G (y, z, −κ 2 )A(Dz , −κ 2 )U (z) dz,
(3.16)
κ > 0, y ∈ Ω, K=I,II,III,IV.
(I)
If U = gj , then (3.16) implies
Z
(I)
(I)
gjj (y, y, −κ 2 ) = − Γj (z − y, −κ 2 ) · P G j (z, y, −κ 2 ) dz S.
(3.17)
S
On the other hand, by the Green formula (1.4) we have
Z
Z
U · A(Dx , −κ 2 )U dx + L[U ] =
U · PU dz S.
Ω
S
(3.18)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
191
(K)
Hence for U = gj we have
Z
(K)
(K)
(K)
gj (z, y, −κ 2 ) · P gj (z, y, −κ 2 ) dz S
L[ gj ] =
(3.19)
S
which on account of (3.17) gives
Z
(I)
(I)
(I)
gj (z, y, −κ 2 ) · P Gj + PΓj dz S =
L[gj ] =
S
Z
(I)
Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S.
= −gjj +
S
(II)
(c) Formula (3.16) implies for gj that
Z (II)
(II)
(II)
gj (y, y, −κ 2 ) =
Gj (y, z, −κ 2 ) · P gj (z, y, −κ 2 ) dz S =
=
Z
S
S
(II)
Gj (y, z, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S.
By relation (3.19), from (3.2) we have
Z
Z
(II) (II)
(II)
(II)
(II)
(II)
L [ gj ] = L[ gj ] − 2
gj · P gj dz S −
gj · PΓj dz S =
S
S
Z
Z
(II)
(II)
gj · PΓj dz S = −
−2
gj · PΓj dz S =
S
=−
Z
S
S
(II)
(II)
Gj + Γj · PΓj dz S = −gjj −
Z
S
Γj · PΓj dz 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
Z
Z
(II) (III)
(III) (III)
(II)
Γj · PΓj dS =
Γj · PΓj dS ≤ L [ gj ] +
−gjj ≤ L [ gj ] +
(III)
S
(III) (I)
= − g ≤ L [g]+
Z
S
S
Γj · PΓj dS =
(I)
−gjj .
By Lemma 3.3 and equalities (3.2) and (3.10), from (3.12) we have
Z
(IV) (I)
′ ′
(IV)
Γj Pj − 2Γ′′j Pj′ − Γj′′ Pj′′ dS =
− gjj ≤ L [gj ] −
Z S
′′ ′
(I)
2Γj (Pj + Pj′′ ) + Γ′j Pj′ − 2Γ′′j Pj′ − Γ′′j Pj′′ dS =
= L[ g j ] −
S
192
M. SVANADZE
(I)
= L[ g j ] −
Z
S
′ ′
(I)
Γj Pj + 2Γ′′j Pj′ ) dS = −gjj .
(K)
Lemma 3.5. The function gjj satisfies the relation
(K)
gjj (y, y, −κ 2 ) < const , y ∈ Ω, δ > 0, K=I,II,III,IV.
ℓ1+δ
y
(3.20)
Proof. First we prove the validity of the formula
Z
2
e j (x − y, −κ 2 ) ≤ g(I)
e j (x − y, −κ 2 ) · A(Dx , −κ 2 )Γ
Γ
jj (y, y, −κ ) dx ≤
B(y,ℓy )
≤
Z
S
Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S,
where B(y, ℓy ) is the ball with center y and radius ℓy .
By property (a) of Lemma 3.2 relation (3.10) implies
Z
(I)
Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S.
gjj (y, y, −κ 2 ) dx ≤
(3.21)
(3.22)
S
e j formula (3.18) gives
On the other hand, for U = Γ
Z
Z
e j dx + L[Γ
ej ] =
e · PΓ
e dS.
e j · A(Dx , −κ 2 )Γ
Γ
Γ
S
Ω
Hence by Lemma 3.1 we have
Z
Z
e j · A(Dx , −κ 2 )Γ
ej ] = −
e j dx +
Γ
L[Γ
Γj · PΓj dS.
Ω
(3.23)
S
By Lemma 3.3 and relation (3.23), from (3.10) we obtain
Z
(I)
(I)
ej ] +
Γj · PΓj dS =
gjj (y, y, −κ 2 ) dx ≥ − L [Γ
Ω
Z
e j dx.
e j · A(Dx , −κ 2 )Γ
=
Γ
(3.24)
B(y,ℓy )
(3.22) and (3.24) imply that (3.21) is valid.
Due to Lemma 3.1 we have
−1
e
Γmj (x − y, −κ 2 ) ≤ const ·r4 ℓ−5
y ≤ const ·ℓy ,
2
e mj (x − y, −κ 2 ) ≤ const ·(xr)2 e−α0 κr r2 ℓ−5 ≤
κ Γ
y
≤ const ·ℓ−3
y , m, j = 1, 6,
2
2
e j (x − y, −κ 2 ) ≤
e
Γj (x − y, −κ ) · A(Dx , −κ )Γ
−4
≤ const ·r2 ℓ−6
y ≤ const ·ℓy .
(3.25)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
193
With (3.25) taken into account, formula (3.21) readily implies
The inequality
(I)
gjj (y, y, −κ 2 ) < const , y ∈ Ω, δ > 0.
ℓ1+δ
y
(3.26)
const
2
g(K)
jj (y, y, −κ ) < 1+δ , y ∈ Ω, δ > 0, K=I,II,III,IV,
ℓy
(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
(K)
Lemma 3.6. The Green tensor G has the property
(K)
(K)
G T (x, y, −κ 2 ) = G (y, x, −κ 2 ),
where x, y ∈ Ω, κ > 0, K=I,II,III,IV.
We introduce the notation
(ρ11 + √ρ0 )I
1
F =p
√
−ρ12 I
ρ11 + ρ22 + 2 ρ0
−ρ12 I
√
,
(ρ22 + ρ0 )I 6×6
(K)
(K)
G = F G F.
It is obvious that the equalities F 2 = E and
(K)
(K)
G T (x, y, −κ 2 ) = G (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(Dx , −κ02 )U (y) = −(ω 2 + κ02 )EU (y),
κ0 > 0,
from (3.16) we obtain
U (x) = (ω 2 + κ02 )
Z
Ω
(K)
G (x, y, −κ02 )EU (y) dy.
(4.1)
194
M. SVANADZE
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
Z (K)
V (x) = (λ∗ + κ02 )
G (x, y, κ02 )V (y) dy,
(4.2)
Ω
∗
2
where V = F U , λ = ω .
One can easily verify [14] that system (4.2) has a discrete spectrum of
(K) (K)
(K)
nonnegative eigenvalues. If ω1 , ω2 , . . . , ωm , . . . are the frequencies of the
eigenoscillations of problem (K) (with their multiplicity taken into account),
(K,1) (K,2)
(K,m)
and U , U , . . . , U , . . . are the corresponding vector-functions, then
(K,m)
(K)
∞
{ ω 2m + κ02 }∞
m=1 and { V }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(K)
(K)
(K)
(K)
(K,m)
∞
sumed that 0 ≤ ω1 ≤ ω2 ≤ . . . , ω1 > 0, limm→∞ ωm = ∞, and { V }m=1
is a system of orthonormalized vectors
Z (K,m)
(K,l)
V (y) · V (y) dy = δml , m, l = 1, 2, 3 . . . .
Ω
(K,m)
∞
will be a system of orthonormalized vector-functions with
Then { U }m=1
weight E, i.e.,
Z
(K,m)
(K,l)
E U (y) · U (y) dy = δml , m, l = 1, 2, 3 . . . , K=I,II,III,IV.
Ω
We have
(K)
Lemma 4.1. The matrix G (x, y, µ0 − κ02 ) is the resolvent of the kernel
(K)
G (x, y, −κ02 ) and the equality
(K)
(K)
G (x, y, −κ 2 ) − G (x, y, −κ02 ) =
Z (K)
(K)
= µ0
G (x, z, −κ02 ) G (z, y, −κ02 ) dz
Ω
is fulfilled, where κ > κ0 > 0, µ0 = κ02 − κ 2 < 0.
Proof. Let
(K)
(K)
(K)
H (x, y, −κ 2 ) = G (x, y, −κ 2 )F − G (x, y, −κ02 )F.
Then for x 6= y the equalities
(K)
(K)
A(Dx , −κ02 ) G (x, y, −κ 2 )F = (κ 2 − κ02 )E G (x, y, −κ 2 )F,
(4.3)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
195
(K)
A(Dx , −κ02 ) G (x, y, −κ02 )F = 0
imply
(K)
(K)
A(Dx , −κ02 ) H (x, y, −κ 2 ) = (κ 2 − κ02 )E G (x, y, −κ 2 )F.
(4.4)
(K)
Moreover, H satisfies the boundary condition
(K)
(K)
B (Dz , n(z)) H (z, y, −κ 2 ) = 0, z ∈ S, y ∈ Ω,
and the equality
(K)
H (x, y, −κ 2 ) = −
+
Z
Z
(K)
Ω
(K)
G (x, z, −κ02 ) · A(Dx , −κ02 ) H (z, y, −κ 2 ) dz +
(K)
(K)
G (x, z, −κ02 ) · P(Dz , n) H (z, y, −κ 2 ) −
S
(K)
T (K)
− P(Dz , n) G (x, z, −κ02 ) · H (x, y, −κ 2 ) dz 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
Z (K)
(K)
(K)
2
2
2
H (x, y, −κ ) = (κ0 − κ ) G (x, z, −κ02 ) · E G (z, y, −κ 2 )F dz. (4.6)
Ω
By multiplying equality (4.6) from the left by the matrix F we obtain relation (4.3).
(K)
(K)
By Lemma 4.1 the matrix G (x, y, −κ 2 ) − G (x, y, −κ02 ) can be decomposed into the series [14]
(K)
(K)
G (x, y, −κ 2 ) − G (x, y, −κ02 ) =
= (κ02 − κ 2 )
(K)
(K)
where λm = ω
(4.7) implies
2
m,
∞
X
(K)
(K)
(K,m)
(K,m)
(λm +κ 2 )−1 (λm +κ02 )−1 V (x)× V (y),
(4.7)
m=1
K=I,II,III,IV, m = 1, 2, 3 . . . , x, y ∈ Ω, κ > κ0 > 0.
(K)
(K)
lim Sp G (x, y, −κ 2 ) − G (x, y, −κ02 ) =
x→y
= (κ02 − κ 2 )
∞
X
m=1
(K)
(K)
(λm + κ 2 )−1 (λm + κ02 )−1 |V (y)|2 .
(4.8)
196
M. SVANADZE
(K)
(K)
Taking into account the equalities Sp G = Sp( G E) and (2.5), from (4.8)
we shall obtain
(K)
(K)
1
(κ0 − κ)M + Sp
g (y, y, −κ 2 ) − g (y, y, −κ02 ) E =
4π
∞ (K)
(K)
(K,m)
X
(4.9)
(λm + κ 2 )−1 (λm + κ02 )−1 | V (y)|2 .
= (κ02 − κ 2 )
m=1
By Lemma 3.5, from (4.9) we have
∞
X
m=1
(K)
(K)
(K,m)
(λm + κ 2 )−1 (λm + κ02 )−1 | V (y)|2 ∼
M
, κ → ∞.
4πκ
(4.10)
By the technique (a Tauber type theorem) used in [17, 19, 21], we obtain
X
(K)
ωm ≤t
(K,m)
| V (y)|2 ∼
M 3
t , t → ∞.
6π 2
(4.11)
By (4.11) we can write the following formula for asymptotic behavior of
eigen vector-functions of problem (K):
X
(K,m)
(K)
ωm ≤t
|F U (y)|2 ∼
M 3
t , t → ∞, K=I,II,III,IV.
6π 2
(4.12)
Integrating equality (4.9) in the domain Ω and applying the fact that the
(K,m)
vectors of the system { V }∞
m=1 are orthonormal, also recalling the results
of Lemma 3.5, we obtain
∞
(K)
X
(K)
−1
1
(λm + κ 2 )(λm + κ02 )
|Ω|M κ −1 + O(κ −2−2δ ),
=
4π
m=1
0
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
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.
177
c 1996 Plenum Publishing Corporation
1072-947X/96/0300-0177$09.50/0
178
M. SVANADZE
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.
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-functions 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 M¨
uller and Niemeyer in [22,
23].
Using V. Avakumovi´c’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-functions.
1. Formulation of Boundary Value Problems
Let the finite domain Ω of the three-dimensional Euclidean space R3
be bounded by the surface S, S ∈ L2 (ε), 0 < ε ≤ 1 [26], 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 = (f1 , f2 , . . . , fk ) and
Pk
ϕ = (ϕ1 , ϕ2 , . . . , ϕk ) will be denoted by f · ϕ =
j=1 fj ϕj , where ϕj is
the complex-conjugate to the number ϕj , f × ϕ = kψml kk×k , ψml = fm ϕl ,
Pk
m, l = 1, k, |f | = ( l=1 fl2 )1/2 . The product of the matrix A = kAml kp×k
Pk
Pk
Pk
by f denotes the vector Af = ( j=1 A1j fj , k=1 A2j fj , . . . , j=1 Apj fj );
The trace of the square matrix
AT is the transposition of the matrix A. P
k
A = kAml kk×k will be denoted by Sp A = j=1 Ajj .
In the absence of mass force, the system of stationary oscillation equations in the shift model of the linear theory of two-component elastic mix-
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
179
tures has the form [3, 5, 6, 8, 27]
a1 ∆u′ + b1 grad div u′ + c∆u′′ + d grad div u′′ −
−α(u′ − u′′ ) + ω 2 ρ11 u′ − ω 2 ρ12 u′′ = 0,
c∆u′ + d grad div u′ + a2 ∆u′′ + b2 grad div u′′ +
(1.1)
+α(u′ − u′′ ) − ω 2 ρ12 u′ + ω 2 ρ22 u′′ = 0,
where u′ = (u′1 , u′2 , u′3 ), u′′ = (u′′1 , u′′2 , u3′′ ) are partial displacements, ∆ is the
three-dimensional Laplace operator, ω is an oscillation frequency, ω > 0,
α ≥ 0,
(−1)j ρ3−j α2
,
ρ1 + ρ2
ρjj = ρj + ρ12 , j = 1, 2, c = µ3 + λ5 ,
ρ2 α2
ρ1 α2
= µ 3 + λ4 − λ5 +
, α2 = λ 3 − λ 4 ,
d = µ3 + λ3 + λ5 −
ρ1 + ρ 2
ρ1 + ρ2
µ1 , µ2 , µ3 , λ1 , λ2 , . . . , λ5 are elastic constants of the mixture [1, 7].
aj = µj − λ5 , bj = µj + λj + λ5 +
In the sequel it will be assumed that the following conditions are fulfilled
[1, 7]:
ρ2 α2
2
+ µ1 > 0, λ5 ≤ 0,
ρ1 + ρ2
3
ρ11 > 0, ρ11 ρ22 > ρ212 ,
ρ1 α2
ρ2 α2
2
2
+ µ1 λ2 +
+ µ2 >
λ1 −
ρ1 + ρ2
3
ρ1 + ρ2
3
2 2
ρ 1 α2
+ µ3 .
> λ3 −
ρ1 + ρ2
3
µ1 > 0, µ1 µ2 > µ23 , λ1 −
System (1.1) can be rewritten in the matrix form as
A(Dx , ω 2 )U (x) ≡ A(Dx ) + αE0 + ω 2 E U (x) = 0,
(1.2)
(1.3)
where
U = (U1 , U2 , . . . , U6 ) = (u′ , u′′ ), x = (x1 , x2 , x3 ) ∈ Ω,
∂
∂
∂
,
,
Dx =
,
∂x1 ∂x2 ∂x3
(1)
A (Dx ) A(2) (Dx )
(j)
A(Dx ) = (2)
, A(j) (Dx ) = kAkl (Dx )k3×3 ,
A (Dx ) A(3) (Dx )6×6
(1)
Akl (Dx ) = a1 ∆δkl + b1
∂2
∂2
(2)
, Akl (Dx ) = c∆δkl + d
,
∂xk ∂xl
∂xk ∂xl
180
M. SVANADZE
∂2
(3)
, j, k, l = 1, 2, 3,
Akl (Dx ) = a2 ∆δkl + b2
∂xk ∂xl
−I I
ρ11 I −ρ12 I
=
E0 =
,
E
, I = kδkl k3×3 ,
I −I
−ρ12 I ρ22 I
6×6
6×6
δkl is the Kronecker symbol.
A vector-function U is called regular in the domain Ω if Uj ∈ C 2 (Ω) ∩
1
C (Ω) (j = 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
(K)
(K)
B (Dz , n(z))U (z) ≡
where
lim
Ω∋x→z∈S
B (Dx , n(z))U (x) = 0,
I
P(Dx , n(z))
(K)
P(Dx , n(z)) + σI
B (Dx , n(z)) =
I
I
(1)
(2)
(3)
(2)
P + P
P +P
for K=I,
for K=II,
for K=III,
,
for K=IV,
6×6
I = kδkl k6×6 , σ > 0, P(Dx , n(z)) is the stress operator in the theory of
elastic mixtures [1],
P (1) P (2)
(j)
P(Dx , n(z) =
, P (j) = kPkl k3×3 ,
P (2) P (3)
6×6
∂
∂
ρ2 α2
∂
(1)
Pkl (Dx , n(z)) = (µ1 −λ5 )δkl
+(µ1 +λ5 )nl
+ λ1 −
,
nk
∂n
∂xk
ρ1 +ρ2
∂xl
∂
ρ1 α2
∂
∂
(2)
+(µ3 −λ5 )nl
+ λ3 −
,
nk
Pkl (Dx , n(z)) = (µ3 +λ5 )δkl
∂n
∂xk
ρ1 +ρ2
∂xl
∂
ρ2 α2
∂
∂
(3)
+(µ2 +λ5 )nl
+ λ2 −
,
Pkl (Dx , n(z)) = (µ2 −λ5 )δkl
nk
∂n
∂xk
ρ1 +ρ2
∂xl
n = (n1 , n2 , n3 ) 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(Dx , −κ 2 )U (x) = 0, x ∈ Ω,
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
181
and the boundary condition
(K)
B (Dz , 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,δ1 , 0 < δ1 ≤ 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]
Z
Z
U (x) · A(Dx )U (x) + W (U, U ) dx = U (z) · P(Dz , n(z))U (z) dz S, (1.4)
S
Ω
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
a = a1 + b1 , b = a2 + b2 , c0 = c + d, d1 = ab − c20 ,
d2 = a1 a2 − c2 , ρ0 = ρ11 ρ22 − ρ212 , ρ3 = ρ11 + ρ22 − 2ρ12 ,
q1 = a + b + 2c0 , q2 = aρ22 + bρ11 + 2c0 ρ12 ,
(1.5)
q3 = a1 + a2 + 2c, q4 = a1 ρ22 + a2 ρ11 + 2cρ12 .
On account of (1.2) we obtain by (1.5)
aj > 0, a > 0, b > 0, dj > 0,
ρ0 > 0, ρ3 > 0, ql > 0, j = 1, 2, l = 1, 4.
Remark 1.1. If α = 0 then (1.1) implies
div u′
2
2
∆ + k 1 ∆ + k2
= 0,
div u′′
rot u′
= 0,
∆ + k32 ∆ + k42
rot u′′
(1.6)
(1.7)
where k12 , k22 and k32 , k42 are the roots of the square equations d1 ξ 2 − ω 2 q2 ξ +
ω 4 ρ0 = 0 and d2 ξ 2 − ω 2 q4 ξ + ω 4 ρ0 = 0, respectively. By virtue of conditions
(1.6) we have kj2 > 0 (j = 1, 4). Assume that kj > 0 (j = 1, 4). By (1.7)
182
M. SVANADZE
it is clear that k1 , k2 and k3 , k4 are the wave numbers of longitudinal and
transverse waves, respectively [3]. Let
cj = ωkj−1 , j = 1, 4.
(1.8)
The constants c1 and c2 will be velocities of longitudinal waves, while c3
and c4 the velocities of transverse waves [3]. Clearly, c21 , c22 and c32 , c24 are
the roots of the equations
ρ0 ξ 2 − q2 ξ + d1 = 0,
2
ρ0 ξ − q4 ξ + d2 = 0.
(1.9)
(1.10)
Remark 1.2. As is well known [28], by the classical theory of elasticity,
in an p
isotropic body one longitudinal wave propagates with the velocity
v1 = p (λ + 2µ)ρ−1 , and two transverse waves with the equal velocities
v2 = µρ−1 (here λ, µ are the Lam´e 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 c1 and c2 , and four transverse waves with
the velocities c3 and c4 (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(Dx , −κ 2 )U (x) = 0
is constructed in [27] in terms of elementary functions. It has the form
(1)
Γ (x, −κ 2 ) Γ(2) (x, −κ 2 )
2
Γ(x, −κ ) = (2)
,
Γ (x, −κ 2 ) Γ(3) (x, −κ 2 )6×6
(2.1)
(j)
Γ(j) = kΓkl k3×3 , j = 1, 2, 3,
n1
(1)
(a2 ∆ − β3 )(∆ − κ12 )(∆ − κ22 )δkl +
Γkl (x, −κ 2 ) =
d2
(1)
(1)
(1)
∂2 o
+ r1 ∆2 + r2 ∆ + r3
γ,
∂xk ∂xl
n 1
(2)
Γkl (x, −κ 2 ) = − (c∆+β2 )(∆−κ12 )(∆−κ22 )δkl +
d2
(2)
(2)
(2)
∂2 o
+ r1 ∆2 + r2 ∆ + r3
γ,
∂xk ∂xl
n1
(3)
(a1 ∆ − β1 )(∆ − κ12 )(∆ − κ22 )δkl +
Γkl (x, −κ 2 ) =
d2
(3)
(3)
(3)
∂2 o
+ r1 ∆2 + r2 ∆ + r3
γ,
∂xk ∂xl
k, l = 1, 2, 3,
(2.2)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
183
where
β1 = α + κ 2 ρ11 , β2 = α + κ 2 ρ12 , β3 = α + κ 2 ρ22 ,
(3)
a2 (2)
c
a
a1
c0
b
− , r1 = − + , r1 =
− ,
d1
d2
d1
d2
d1
d2
(1)
1
r2 =
2β2 (a2 d − cb2 ) + β3 (2a2 b1 − 2cd − b1 b2 − d2 ) ,
d1 d 2
(2)
1
r2 =
β1 (a1 d − b2 c) + β2 (a2 b1 − 2cd + ab2 − d2 ) + β3 (a1 d − b1 c) ,
d1 d 2
(3)
1
r2 =
β1 (2a1 b2 − 2cd + b1 b2 − d2 ) + 2β2 (a1 d − b1 c) ,
d1 d 2
(1)
1 2
r3 = −
b1 β3 + 2dβ2 β3 + b2 β22 ,
d1 d2
(2)
1
b1 β2 β3 + b2 β1 β2 + dβ1 β3 + dβ22 ,
r3 = −
d1 d2
4
X
(3)
1 2
2
r3 = −
ηj γj ,
bβ2 + 2dβ1 β2 + b2 β1 , γ =
d1 d2
j=1
(1)
r1
=
ηj =
4
Y
l=1
l6=j
(κj2 − κl2 )−1 , γj (x, −κ 2 ) = −
e−κj |x|
, j = 1, 4,
4π|x|
κ12 , κ22 and κ32 , κ42 are, respectively, the roots of the following square equations
d1 ξ 2 − (aβ3 + bβ1 + 2c0 β2 )ξ + β1 β3 − β22 = 0
and
d2 ξ 2 − (a1 β3 + a2 β1 + 2cβ2 )ξ + β1 β3 − β22 = 0.
By virtue of (1.6) we have κj2 > 0 (j = 1, 4). Assume that κj > 0 (j = 1, 4)
and introduce the notation
−1
−1 (0)
(0)
2
2
) ,
) , ql = d2 κl2 (κl2 −κ7−l
qj = d1 κj2 (κj2 −κ3−j
(1)
qj
(1)
ql
(0)
(2)
(0)
(3)
(0)
= (bκj2 −β3 ) qj , qj = −(c0 κj2 +β2 ) qj , qj = (aκj2 −β1 ) qj ,
(0)
(2)
(0)
(3)
(0)
= (a2 κj2 −β3 ) ql , ql = −(cκl2 +β2 ) ql , ql = (a1 κl2 −β1 ) ql ,
(1)
(2)
(3)
pk = ρ11 qk −2ρ12 qk +ρ22 qk , j = 1, 2, l = 3, 4, k = 1, 4.
We have
Lemma 2.1. The matrix Γ(x, −κ 2 ) has the following properties:
(a) ΓT (x, −κ 2 ) = Γ(x, −κ 2 );
(b) Γ(−x, −κ 2 ) = Γ(x, −κ 2 );
(2.3)
184
M. SVANADZE
(c) A(Dx , −κ 2 )Γ(x, −κ 2 ) = δ(x)I, where δ(x) is the Dirac distribution
function;
(d) for x 6= 0 the matrices Γ(1) , Γ(2) , Γ(3) have the form
Γ(k) (x, −κ 2 ) = grad div
− rot rot
2
X
(k)
qj γj (x, −κ 2 ) −
j=1
4
X
l=3
(k)
ql γl (x, −κ 2 ),
k = 1, 2, 3;
(2.4)
√
1
Γ(x, −κ 2 ) − Γ(x, −κ02 ) E =
(2.5)
κM + O( κ),
x→0
4π
κ > κ0 > 0, κ → ∞;
s
∂
c∗ e−α0 κ|x|
(f) s1 s2 s3 Γkl (x, −κ 2 ) <
,
∂x1 ∂x2 ∂x3
|x|1+s
where c∗ = const > 0, s = s1 + s2 + s3 , s1 , s2 , s3 are nonnegative integer
numbers, α0 is a positive number not depending on κ and x,
(e)
lim Sp
−3
−3
−3
M = c−3
1 + c2 + 2(c3 + c4 ).
(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
∆γj (x, −κ 2 ) = κj2 γj (x, −κ 2 ),
1
Iγj (x, −κ 2 ) = 2 grad div − rot rot γj (x, −κ 2 ), x 6= 0, j = 1, 4,
κj
we find by (2.2) that
nh 1
2
2
2
2
2
2 (a2 κj − β3 )(κj − κ1 )(κj − κ2 ) +
d
κ
2
j
j=1
i
(1)
(1)
(1)
+ r1 κj4 + r2 κj2 + r3 grad div −
o
(a2 κj2 − β3 )(κj2 − κ12 )(κj2 − κ22 ) rot rot γj (x, −κ 2 ). (2.7)
Γ(1) (x, −κ 2 ) =
−
1
d2 κj2
4
X
ηj
Using the relations
(1)
(1)
(1)
1
(a2 κj2 − β3 )(κj2 − κ12 )(κj2 − κ22 ) + r1 κj4 + r2 κj2 + r3 =
d2 κj2
1
=
(bκj2 − β3 )(κj2 − κ32 )(κj2 − κ42 ), j = 1, 4,
d1 κj2
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
185
which are easy to verify, from (2.7) we obtain
h 1
2
2
2
2
2
2 (bκj − β3 )(κj − κ3 )(κj − κ4 ) grad div −
d
κ
1
j
j=1
i
(a2 κj2 − β3 )(κj2 − κ12 )(κj2 − κ22 ) rot rot γj (x, −κ 2 ) =
Γ(1) (x, −κ 2 ) =
4
X
ηj
−
1
d2 κj2
=
2
X
j=1
1
(bκj2 − β3 )(κj2 − κ32 )(κj2 − κ42 )ηj grad div γj (x, −κ 2 ) −
d1 κj2
4
X
1
(a2 κj2 − β3 )(κl2 − κ12 )(κl2 − κ22 )ηl rot rot γl (x, −κ 2 ) =
d2 κj2
−
l=3
= grad div
2
4
X
X
(1)
(1)
ql γl (x, −κ 2 ).
qj γj (x, −κ 2 ) − rot rot
j=1
l=3
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
(1)
Φ
Φ(2)
(l)
Φ = (3)
, Φ(l) = kΦkj k3×3 , l = 1, 4,
(4)
Φ
Φ
6×6
Φ(1) = ρ11 Γ(1) − ρ12 Γ(2) ,
Φ(3) = ρ11 Γ(2) − ρ12 Γ(3) ,
Φ(2) = −ρ12 Γ(1) + ρ22 Γ(2) ,
Φ(4) = −ρ12 Γ(2) + ρ22 Γ(3) .
Clearly, Sp Φ = Sp(ρ11 Γ(1) − 2ρ12 Γ(2) + ρ22 Γ(3) ). Hence on account of (2.3),
(2.4) we obtain
4
2
i
h
X
X
Sp Φ(x, −κ 2 ) = Sp grad div
pl γl (x, −κ 2 ) =
pj γj (x, −κ 2 ) − rot rot
j=1
=∆
2
hX
j=1
=
2
X
j=1
Using the equalities
have
l=3
pj γj (x, −κ 2 ) + 2
pj κj2 γj (x, −κ 2 ) + 2
P2
j=1
l=3
4
X
l=3
pj κj2 = q2 d−1
1 ,
i
pl γl (x, −κ 2 ) =
pl κl2 γl (x, −κ 2 ).
P4
l=3
(2.8)
pl κl2 = q4 d−1
2 , from (2.8) we
i
X
1 hX
pj κj2 + 2
pl κl2 − p5 (κ)|x| +
4π|x| j=1
2
Sp Φ(x, −κ 2 ) = −
4
X
4
l=3
186
M. SVANADZE
i
2q4
1 h q2
+
− p5 (κ)|x| + O(|x|), |x| ≪ 1, (2.9)
4π|x| d1
d2
P2
P4
where p5 (κ) = j=1 pj κj3 + 2 l=3 pl κl3 . The relation (2.9) readily implies
+O(|x|) = −
1
p5 (κ) − p5 (κ0 ) , κ0 > 0. (2.10)
lim Sp Φ(x, −κ 2 ) − Φ(x, −κ02 ) =
x→0
4π
Let us now calculate the difference p5 (κ) − p5 (κ0 ) for κ > κ0 > 0,
κ → ∞. Assume that τ12 , τ22 and τ32 , τ42 are the roots of the square equations
d1 ξ 2 − κ 2 q2 ξ + κ 4 ρ0 = 0 and dξ 2 − κ 2 q4 ξ + κ 4 ρ0 = 0, respectively. By (1.6)
we have τj2 > 0 (j = 1, 4). Assuming that τj > 0 (j = 1, 4), we obtain
(j = 1, 4), where cj (j = 1, 4) are defined by (1.8). Taking into
τj = κc−1
j
account the relations
τ12 + τ22 = κ 2 q2 d1−1 , τ12 τ22 = κ 4 ρ0 d−1
1 ,
κ12 + κ1 κ2 + κ22 = τ12 + τ1 τ2 + τ22 + O(κ),
i
τ1 + τ2 = (κ1 + κ2 ) 1 + O(κ −1/2 ) , κ → ∞,
we find by (2.3) that
2
X
pj κj3 =
j=1
hq
αρ3 i
1
2κ 2 ρ0
2
(κ12 + κ1 κ2 + κ22 ) −
−
=
κ1 + κ2 d1
d1
d1
αρ3
1 τ1 + τ2
(τ 3 + τ23 ) +
=
κ 2 κ1 + κ2 1
d1 (κ1 + κ2 )
√
= κ(c1−3 + c2−3 ) + O( κ), κ → ∞.
=
Similarly, one can show that
Therefore the equality
P4
l=3
√
−3
pl κl2 = κ(c−3
3 +c4 )+O( x), x → ∞.
√
p5 (κ) − p5 (κ0 ) = κM + O( κ), κ > κ0 > 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
(K)
(K)
G (x, y, −κ 2 ) = −Γ(x − y, −κ 2 ) + g (x, y, −κ 2 )
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
187
will be called the Green tensor of problem (K) (K=I, II, III, IV) if the
(K)
matrix g satisfies the homogeneous equation
(K)
A(Dx , −κ 2 ) g (z, y, −κ 2 ) = 0, x, y ∈ Ω,
and the boundary condition
(K)
(K)
(K)
B (Dz , n(z)) g (z, y, −κ 2 ) = B (Dz , n(z))Γ(z − y, −κ 2 ), z ∈ S, y ∈ Ω.
(K)
The matrix g (K=I, II, III, IV) is a solution of problem (K)∗f with
(K)
(K)
a special boundary value f = B Γ. The existence and uniqueness of g
follows from Lemma 1.1.
e
e = (e
u′ , u
e′′ ). Assume that U, U
e′′2 , u
e′′3 ), U
e′′ = (e
Let u
u′′1 , u
u′1 , u
e′2 , u
e′ = (e
e′3 ), u
are the vector-functions with real components. Introduce the notation [1, 7]
ε′lj =
εe′lj =
hlj =
e
hlj =
e) =
W (U, U
+
+
1 ∂u′j
1 ∂uj′′
∂u′l ′′
∂u′′l
+
+
, εlj =
,
2 ∂xl
∂xj
2 ∂xl
∂xj
u′j
u′′j
1 ∂e
∂e
ul′ ′
∂e
u′′l
1 ∂e
+
+
, εelj =
,
2 ∂xl
∂xj
2 ∂xl
∂xj
∂uj′′
∂u′l
∂u′′l
1 ∂u′j
−
+
−
,
2 ∂xl
∂xj
∂xl
∂xj
u′j
u′′j
∂e
∂e
ul′
∂e
u′′l
1 ∂e
, l, j = 1, 2, 3,
−
+
−
2 ∂xl
∂xj
∂xl
∂xj
3 h
X
′
′′ ′
(λ1 εll
+ λ3 ε′′ll )e
ε′jj + 2µ1 ε′lj εe′lj + 2µ3 εlj
εelj +
l,j=1
(λ4 ε′ll
′′
′′ ′′
+
εjj + 2µ3 ε′lj εe′′lj + 2µ2 ε′′lj εelj
)e
+ λ2 εll
i
α2
′′
− εe′jj ) − 2λ5 hlj e
hlj .
εjj
(ρ2 ε′ll + ρ1 ε′′ll )(e
ρ1 + ρ2
Consider the functions
Z
e dx,
e]=
e + κ2 U E U
e )−αU E0 U
L[U, U
W (U, U
Ω
R
e +κ 2 U E U
e ]dx ≡ L[U ] for
[W
U
(U,
)−αU E0 U
Ω
R
L[U ]−2 S U (z) · PΓj (z − y, −κ 2 )dz S
for
(K)
L [U ] = (II)
R
L [U ]+σ [U (z)−Γ (z − y, −κ 2 )]2 d S
for
j
z
R S
e j (z − y, −κ 2 )dz S
L[U ]−2 S u′′ (z) · PΓ
for
(3.1)
K=I,
K=II,
K=III,
K=IV,
(3.2)
188
M. SVANADZE
where Γj is the jth column of the matrix Γ, j is some fixed number, j = 1, 6,
e = kP (1) + P (2) , P (2) + P (3) k6×6 , σ > 0.
P
It is assumed that
(K)
R = U : U ∈ C 2 (Ω) ∩ C 1 (Ω), B U (z) =
(K)
(K)
= B Γj (z − y, −κ 2 ), z ∈ S, y ∈ Ω
(3.3)
(K)
is the set of admissible vector-functions for the functional L (K=I,II,III,IV).
Let [17]
n
h
r m il o
e − y, −κ 2 ) = 1 − 1 −
Γ(x
Γ(x − y, −κ 2 ),
(3.4)
ρy (x)
where r = |x − y|, ρy (x) = max{r, ℓy }, ℓy is the distance from the point y
to S, while m, l are natural numbers.
The following lemmas hold.
e has the properties:
Lemma 3.1. The matrix Γ
(K)
e j ∈ R , j = 1, 6, K=I,II,III,IV;
a) Γ
e j (x − y, −κ 2 ) = Γj (x − y, −κ 2 ) for r ≥ ℓy ;
b) Γ
∂s
e kj (x − y, −κ 2 ) < const e−α0 κr rm−s−1 for r < ℓy ,
c) s1 s2 s3 Γ
∂x1 ∂x2 ∂x3
ℓm
y
m ≥ s + 1, l ≥ s + 1, s = s1 + s2 + s3 , s1 , s2 , s3 are negative numbers, α0
e j is the jth column of
is a positive number not depending on κ, x, y, and Γ
e
the matrix Γ, j, k = 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).
e are vector-functions, regular in Ω, then the
Lemma 3.2. If U and U
functionals L and L have the properties:
(a) L[U ] ≥ 0. If L[U ] = 0, then U (x) ≡ 0, x ∈ Ω;
e ] = L[U
e , U ];
(b) L[U, U
(c) L[U, U ] = L[U ];
e
e
e
(d) L[U
Z + U ] = L[U ] + 2L[U, U ] + L[U ]; Z
e · PU dS;
e · A(Dx , −κ 2 )U dx + L[U, U
e] =
U
(e)
(3.5)
U
S
Ω
Z
e · P (K)
e , (K)
U
gj dS, K=I,II,III,IV;
(3.6)
gj ] =
(f) L[U
S
(I)
e , (I)
e = U − (I)
gj ] = 0 for U
(g) L[U
gj , U ∈ R, j = 1, 6.
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
189
Proof. The validity of properties (a), (b), (c), (d) is obvious by (3.1). Relation (3.5) follows from the Green formula [1]
Z
Z
e · PU dS.
e
e
U
U · A(Dx )U + W (U, U ) dx =
Ω
S
(I)
(K)
If U = gj , then from (3.5) we obtain equality (3.6). If U ∈ R, then
e (z) = 0, z ∈ S.
e = U − (I)
the vector U
gj satisfies the boundary condition U
e , (I)
gj ] = 0.
Therefore (3.6) implies L[U
(K)
Lemma 3.3. The functional L has a minimum value only on the vector
(K)
gj ,
i.e.,
(K)
(K) (K)
min L [U ] = L [ gj ], K =I,II,III,IV.
(3.7)
(K)
U∈ R
(I)
e ≡ U − (I)
gj , then by Lemma 3.2 we have
Proof. If U ∈ R, U
(I)
(I)
(I)
e ] = L[gj ] + 2L[ g j , U
e ] + L[U
e] =
L[U ] = [L[gj + U
(III)
(I)
e ] ≥ L[(I)
= L[gj ] + L[U
gj ].
(III)
e = U − gj . Then
Let U ∈ R , gj = gj , U
Z
Z
(III)
e
e
e · PΓj dS +
L [U ] = L[gj ] + 2L[gj , U ]+L[U ]−2 gj PΓj dS −2 U
S
S
Z
Z
Z
e 2 dS.
e dS +σ
+σ [gj −Γj ]2 dS + 2σ (gj −Γj ) · U
(3.8)
U
S
S
S
By virtue of (3.2), (3.6) and the boundary condition Pgj + σgj = PΓj +
σΓj we find from (3.8) that
Z
(III)
(III)
(III)
e] + σ
e 2 dS ≥ L [gj ].
L [U ] ≡ L [gj ] + L[U
U
S
Quite similarly, equality (3.7) is proved for K=II,IV. Property (a) of
(K)
Lemma 3.2 implies that the functional in the set R has a minimum only
(K)
on gj (K=I,II,III,IV).
(I) (II) (III) (IV)
Lemma 3.4. The vector-functions gj , gj , gj , gj satisfy the following
relations:
190
M. SVANADZE
(II) (III)
L [ gj ]
(a)
(b)
(III) (III)
≤ L [ gj ],
(3.9)
(I)
(I)
gjj (y, y, −κ 2 ) = −L[gj ]+
Z
+ Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S;
(3.10)
S
(c)
(K) (K)
(K)
gjj (y, y, −κ 2 ) = − L [ gj ]−
Z
− Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S, K=II,III;
(3.11)
S
(IV) (IV)
(IV)
gjj (y, y, −κ 2 ) = − L [ gj ]+
(d)
(e)
Z
[Γ′j Pj′ −2Γ′′j Pj′ −Γ′′j Pj′′ ] dz S;
(3.12)
S
(III) (I)
(I)
gjj (y, y, −κ 2 ) = − L [gj ]−
Z
− Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S;
(3.13)
S
(I)
(III)
(II)
gjj (y, y, −κ 2 ) ≤ gjj (y, y, −κ 2 ) ≤ gjj (y, y, −κ 2 ),
(f)
(IV)
(I)
gjj (y, y, −κ 2 ) ≤ gjj (y, y, −κ 2 ),
(3.14)
(3.15)
where y ∈ Ω, Γ′j = (Γ1j , Γ2j , Γ3j ), Γ′′j = (Γ4j , Γ5j , Γ6j ), Pj′ = (P (1) +
P (2) )Γj′ , Pj′′ = (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
Z (K)
(K)
T
G (y, z, −κ 2 )PU (z) − P G (z, y, −κ 2 ) U (z) dz S −
U (y) =
−
Z
S
(K)
Ω
G (y, z, −κ 2 )A(Dz , −κ 2 )U (z) dz,
(3.16)
κ > 0, y ∈ Ω, K=I,II,III,IV.
(I)
If U = gj , then (3.16) implies
Z
(I)
(I)
gjj (y, y, −κ 2 ) = − Γj (z − y, −κ 2 ) · P G j (z, y, −κ 2 ) dz S.
(3.17)
S
On the other hand, by the Green formula (1.4) we have
Z
Z
U · A(Dx , −κ 2 )U dx + L[U ] =
U · PU dz S.
Ω
S
(3.18)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
191
(K)
Hence for U = gj we have
Z
(K)
(K)
(K)
gj (z, y, −κ 2 ) · P gj (z, y, −κ 2 ) dz S
L[ gj ] =
(3.19)
S
which on account of (3.17) gives
Z
(I)
(I)
(I)
gj (z, y, −κ 2 ) · P Gj + PΓj dz S =
L[gj ] =
S
Z
(I)
Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S.
= −gjj +
S
(II)
(c) Formula (3.16) implies for gj that
Z (II)
(II)
(II)
gj (y, y, −κ 2 ) =
Gj (y, z, −κ 2 ) · P gj (z, y, −κ 2 ) dz S =
=
Z
S
S
(II)
Gj (y, z, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S.
By relation (3.19), from (3.2) we have
Z
Z
(II) (II)
(II)
(II)
(II)
(II)
L [ gj ] = L[ gj ] − 2
gj · P gj dz S −
gj · PΓj dz S =
S
S
Z
Z
(II)
(II)
gj · PΓj dz S = −
−2
gj · PΓj dz S =
S
=−
Z
S
S
(II)
(II)
Gj + Γj · PΓj dz S = −gjj −
Z
S
Γj · PΓj dz 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
Z
Z
(II) (III)
(III) (III)
(II)
Γj · PΓj dS =
Γj · PΓj dS ≤ L [ gj ] +
−gjj ≤ L [ gj ] +
(III)
S
(III) (I)
= − g ≤ L [g]+
Z
S
S
Γj · PΓj dS =
(I)
−gjj .
By Lemma 3.3 and equalities (3.2) and (3.10), from (3.12) we have
Z
(IV) (I)
′ ′
(IV)
Γj Pj − 2Γ′′j Pj′ − Γj′′ Pj′′ dS =
− gjj ≤ L [gj ] −
Z S
′′ ′
(I)
2Γj (Pj + Pj′′ ) + Γ′j Pj′ − 2Γ′′j Pj′ − Γ′′j Pj′′ dS =
= L[ g j ] −
S
192
M. SVANADZE
(I)
= L[ g j ] −
Z
S
′ ′
(I)
Γj Pj + 2Γ′′j Pj′ ) dS = −gjj .
(K)
Lemma 3.5. The function gjj satisfies the relation
(K)
gjj (y, y, −κ 2 ) < const , y ∈ Ω, δ > 0, K=I,II,III,IV.
ℓ1+δ
y
(3.20)
Proof. First we prove the validity of the formula
Z
2
e j (x − y, −κ 2 ) ≤ g(I)
e j (x − y, −κ 2 ) · A(Dx , −κ 2 )Γ
Γ
jj (y, y, −κ ) dx ≤
B(y,ℓy )
≤
Z
S
Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S,
where B(y, ℓy ) is the ball with center y and radius ℓy .
By property (a) of Lemma 3.2 relation (3.10) implies
Z
(I)
Γj (z − y, −κ 2 ) · PΓj (z − y, −κ 2 ) dz S.
gjj (y, y, −κ 2 ) dx ≤
(3.21)
(3.22)
S
e j formula (3.18) gives
On the other hand, for U = Γ
Z
Z
e j dx + L[Γ
ej ] =
e · PΓ
e dS.
e j · A(Dx , −κ 2 )Γ
Γ
Γ
S
Ω
Hence by Lemma 3.1 we have
Z
Z
e j · A(Dx , −κ 2 )Γ
ej ] = −
e j dx +
Γ
L[Γ
Γj · PΓj dS.
Ω
(3.23)
S
By Lemma 3.3 and relation (3.23), from (3.10) we obtain
Z
(I)
(I)
ej ] +
Γj · PΓj dS =
gjj (y, y, −κ 2 ) dx ≥ − L [Γ
Ω
Z
e j dx.
e j · A(Dx , −κ 2 )Γ
=
Γ
(3.24)
B(y,ℓy )
(3.22) and (3.24) imply that (3.21) is valid.
Due to Lemma 3.1 we have
−1
e
Γmj (x − y, −κ 2 ) ≤ const ·r4 ℓ−5
y ≤ const ·ℓy ,
2
e mj (x − y, −κ 2 ) ≤ const ·(xr)2 e−α0 κr r2 ℓ−5 ≤
κ Γ
y
≤ const ·ℓ−3
y , m, j = 1, 6,
2
2
e j (x − y, −κ 2 ) ≤
e
Γj (x − y, −κ ) · A(Dx , −κ )Γ
−4
≤ const ·r2 ℓ−6
y ≤ const ·ℓy .
(3.25)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
193
With (3.25) taken into account, formula (3.21) readily implies
The inequality
(I)
gjj (y, y, −κ 2 ) < const , y ∈ Ω, δ > 0.
ℓ1+δ
y
(3.26)
const
2
g(K)
jj (y, y, −κ ) < 1+δ , y ∈ Ω, δ > 0, K=I,II,III,IV,
ℓy
(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
(K)
Lemma 3.6. The Green tensor G has the property
(K)
(K)
G T (x, y, −κ 2 ) = G (y, x, −κ 2 ),
where x, y ∈ Ω, κ > 0, K=I,II,III,IV.
We introduce the notation
(ρ11 + √ρ0 )I
1
F =p
√
−ρ12 I
ρ11 + ρ22 + 2 ρ0
−ρ12 I
√
,
(ρ22 + ρ0 )I 6×6
(K)
(K)
G = F G F.
It is obvious that the equalities F 2 = E and
(K)
(K)
G T (x, y, −κ 2 ) = G (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(Dx , −κ02 )U (y) = −(ω 2 + κ02 )EU (y),
κ0 > 0,
from (3.16) we obtain
U (x) = (ω 2 + κ02 )
Z
Ω
(K)
G (x, y, −κ02 )EU (y) dy.
(4.1)
194
M. SVANADZE
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
Z (K)
V (x) = (λ∗ + κ02 )
G (x, y, κ02 )V (y) dy,
(4.2)
Ω
∗
2
where V = F U , λ = ω .
One can easily verify [14] that system (4.2) has a discrete spectrum of
(K) (K)
(K)
nonnegative eigenvalues. If ω1 , ω2 , . . . , ωm , . . . are the frequencies of the
eigenoscillations of problem (K) (with their multiplicity taken into account),
(K,1) (K,2)
(K,m)
and U , U , . . . , U , . . . are the corresponding vector-functions, then
(K,m)
(K)
∞
{ ω 2m + κ02 }∞
m=1 and { V }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(K)
(K)
(K)
(K)
(K,m)
∞
sumed that 0 ≤ ω1 ≤ ω2 ≤ . . . , ω1 > 0, limm→∞ ωm = ∞, and { V }m=1
is a system of orthonormalized vectors
Z (K,m)
(K,l)
V (y) · V (y) dy = δml , m, l = 1, 2, 3 . . . .
Ω
(K,m)
∞
will be a system of orthonormalized vector-functions with
Then { U }m=1
weight E, i.e.,
Z
(K,m)
(K,l)
E U (y) · U (y) dy = δml , m, l = 1, 2, 3 . . . , K=I,II,III,IV.
Ω
We have
(K)
Lemma 4.1. The matrix G (x, y, µ0 − κ02 ) is the resolvent of the kernel
(K)
G (x, y, −κ02 ) and the equality
(K)
(K)
G (x, y, −κ 2 ) − G (x, y, −κ02 ) =
Z (K)
(K)
= µ0
G (x, z, −κ02 ) G (z, y, −κ02 ) dz
Ω
is fulfilled, where κ > κ0 > 0, µ0 = κ02 − κ 2 < 0.
Proof. Let
(K)
(K)
(K)
H (x, y, −κ 2 ) = G (x, y, −κ 2 )F − G (x, y, −κ02 )F.
Then for x 6= y the equalities
(K)
(K)
A(Dx , −κ02 ) G (x, y, −κ 2 )F = (κ 2 − κ02 )E G (x, y, −κ 2 )F,
(4.3)
ASYMPTOTIC BEHAVIOR OF EIGENFUNCTIONS
195
(K)
A(Dx , −κ02 ) G (x, y, −κ02 )F = 0
imply
(K)
(K)
A(Dx , −κ02 ) H (x, y, −κ 2 ) = (κ 2 − κ02 )E G (x, y, −κ 2 )F.
(4.4)
(K)
Moreover, H satisfies the boundary condition
(K)
(K)
B (Dz , n(z)) H (z, y, −κ 2 ) = 0, z ∈ S, y ∈ Ω,
and the equality
(K)
H (x, y, −κ 2 ) = −
+
Z
Z
(K)
Ω
(K)
G (x, z, −κ02 ) · A(Dx , −κ02 ) H (z, y, −κ 2 ) dz +
(K)
(K)
G (x, z, −κ02 ) · P(Dz , n) H (z, y, −κ 2 ) −
S
(K)
T (K)
− P(Dz , n) G (x, z, −κ02 ) · H (x, y, −κ 2 ) dz 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
Z (K)
(K)
(K)
2
2
2
H (x, y, −κ ) = (κ0 − κ ) G (x, z, −κ02 ) · E G (z, y, −κ 2 )F dz. (4.6)
Ω
By multiplying equality (4.6) from the left by the matrix F we obtain relation (4.3).
(K)
(K)
By Lemma 4.1 the matrix G (x, y, −κ 2 ) − G (x, y, −κ02 ) can be decomposed into the series [14]
(K)
(K)
G (x, y, −κ 2 ) − G (x, y, −κ02 ) =
= (κ02 − κ 2 )
(K)
(K)
where λm = ω
(4.7) implies
2
m,
∞
X
(K)
(K)
(K,m)
(K,m)
(λm +κ 2 )−1 (λm +κ02 )−1 V (x)× V (y),
(4.7)
m=1
K=I,II,III,IV, m = 1, 2, 3 . . . , x, y ∈ Ω, κ > κ0 > 0.
(K)
(K)
lim Sp G (x, y, −κ 2 ) − G (x, y, −κ02 ) =
x→y
= (κ02 − κ 2 )
∞
X
m=1
(K)
(K)
(λm + κ 2 )−1 (λm + κ02 )−1 |V (y)|2 .
(4.8)
196
M. SVANADZE
(K)
(K)
Taking into account the equalities Sp G = Sp( G E) and (2.5), from (4.8)
we shall obtain
(K)
(K)
1
(κ0 − κ)M + Sp
g (y, y, −κ 2 ) − g (y, y, −κ02 ) E =
4π
∞ (K)
(K)
(K,m)
X
(4.9)
(λm + κ 2 )−1 (λm + κ02 )−1 | V (y)|2 .
= (κ02 − κ 2 )
m=1
By Lemma 3.5, from (4.9) we have
∞
X
m=1
(K)
(K)
(K,m)
(λm + κ 2 )−1 (λm + κ02 )−1 | V (y)|2 ∼
M
, κ → ∞.
4πκ
(4.10)
By the technique (a Tauber type theorem) used in [17, 19, 21], we obtain
X
(K)
ωm ≤t
(K,m)
| V (y)|2 ∼
M 3
t , t → ∞.
6π 2
(4.11)
By (4.11) we can write the following formula for asymptotic behavior of
eigen vector-functions of problem (K):
X
(K,m)
(K)
ωm ≤t
|F U (y)|2 ∼
M 3
t , t → ∞, K=I,II,III,IV.
6π 2
(4.12)
Integrating equality (4.9) in the domain Ω and applying the fact that the
(K,m)
vectors of the system { V }∞
m=1 are orthonormal, also recalling the results
of Lemma 3.5, we obtain
∞
(K)
X
(K)
−1
1
(λm + κ 2 )(λm + κ02 )
|Ω|M κ −1 + O(κ −2−2δ ),
=
4π
m=1
0