GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 2, 1999, 107-126
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
AND EIGENVALUES OF THE BASIC
BOUNDARY-CONTACT OSCILLATION PROBLEMS OF
THE CLASSICAL THEORY OF ELASTICITY
T. BURCHULADZE AND R. RUKHADZE
Abstract. The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman’s
method, the asymptotic formulas for the distribution of eigenfunctions and eigenvalues are obtained.
1. After the remarkable papers of T. Carleman [1–2] the method based
on the asymptotic investigation of the resolvent kernel (or of any other
function of the considered operator) with a subsequenet use of Tauberian
theorems has become quite popular. By generalizing Carleman’s method
(and combining it with the variational one) A. Plejel [3] derived the asymptotic formulas for the distribution of eigenfunctions and eigenvalues of the
boundary value oscillation problems of classical elasticity. Mention should
also be made of T. Burchuladze’s papers [4–5], where the asymptotic formulas for the distribution of eigenfunctions of the boundary value oscillation
problems are obtained for isotropic and anisotropic elastic bodies using integral equations and Carleman’s method. Further progress in this direction
was made by R. Dikhamindzhia [6]. He obtained the asymptotic formulas
for the distribution of eigenfunctions and eigenvalues for two- and threedimensional boundary value oscillation problems of couple-stress elasticity
which generalize analogous formulas of classical elasticity. In his recent work
M. Svanadze [7] derived the asymptotic formulas for oscillation boundary
value problems of the linear theory of mixtures of two homogeneous isotropic
elastic materials.
1991 Mathematics Subject Classification. 73C02.
Key words and phrases. Carleman method, boundary-contact oscillation problems of
elasticity, eigenfunctions and eigenvalues, asymptotic formulas for distribution, Green’s
tensor.
107
c 1999 Plenum Publishing Corporation
1072-947X/99/0300-0107$15.00/0 
108
T. BURCHULADZE AND R. RUKHADZE
2. Throughout the paper we shall use the following notation: x =
€ P3

2 1/2
(x1 , x2 , x3 ), y = (y1 , y2 , y3 ) are points of R3 ; |x − y| =
k=1 (xk − yk )
is the distance between the points x and y; D0 ⊂ R3 is a finite domain
bounded by closed surfaces S0 , S1 , . . . , Sm of the class Ω2 (α) 0 < α ≤ 1
[8] with S0 covering all other Sk while the latter surfaces not covering each
6 k, i, k = 0, m; the finite domain bounded by Sk
other; Si ∩ Sk = ∅ for i =

€ m
(k = 1, m) will be denoted by Dk ; D0 = D0 ∪ ∪ Sk , Dk = Dk ∪ Sk ,
k=0
k = 1, m.
If u and v are the three-component real vectors u = (u1 , u2 , u3 ) and v =
P3
(v1 , v2 , v3 ), then uv = i=1 ui vi is the scalar product of these vectors; |u| =

€ P3
2 1/2
. The matrix product is obtained by multipliing a row by a
i=1 ui
column; the sign [·]T denotes the operation of transposition; if A = kAij k3×3
P3
2
. Any vector u = (u1 , u2 , u3 ) is
is a 3 × 3 matrix, then |A|2 = i,j=1 Aij
treated as a 3 × 1 one-column matrix: u = kui k3×1 ; Ak = kAjk k3j=1 is the
k-th column vector of the matrix A.
The vector u = (u1 , u2 , u3 ) will be called regular in Dk if
ui ∈ C 1 (Dk ) ∩ C 2 (Dk ), i = 1, 2, 3.
A system of differential homogeneous equations of oscillation of classical
elasticity for a homogeneous isotropic elastic medium has the form [8]
µ∆u + (λ + µ) grad div u + ρω 2 u = 0,
(1)
where u(x) = (u1 , u2 , u3 ) is the displacement vector, ∆ is the three-dimensional Laplace operator, ρ = const > 0 is the medium density, ω is the
oscillation frequency, λ and µ are the elastic Lamé constants satisfying the
natural conditions
µ > 0, 3λ + 2µ > 0.
We introduce the matrix differential operator


A(∂x) = Aij (∂x)3×3 ,
Aij (∂x ) = δij µρ−1 ∆ + (λ + µ)ρ−1
∂2
,
∂xi ∂xj
where δij is the Kronecker symbol. Then equation (1) can be rewritten in
the vector-matrix form
A(∂x )u + ω 2 u = 0.
The matrix-differential operator
where


T (∂x , n(x)) = Tij (∂x , n(x))3×3 ,
Tij (∂x , n(x)) = λρ−1 ni (x)
∂
∂
∂
+ λρ−1 nj (x)
+ µρ−1 δij
,
∂xj
∂xi
∂n(x)
(2)
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
109
n(x) is an arbitrary unit vector at the point x (if x ∈ Sk , then k = 0, m
is the normal unit vector external with respect to the domain D0 ) is called
the stress operator.
It will be assumed that the domains Dk , k = 0, m0 are filled with homogeneous isotropic elastic media with the Lamé constants λk , µk and density
ρk , while the other domains Dk , k = m0 + 1, m are hollow inclusions. When
the operators A and T contain λk and µk instead of λ and µ, we shall write
k
k
A and T , respectively.
We introduce the notation
u+ (z) =
lim
D0 3x→z∈Sk
u(x), k = 0, m; u− (z) =
lim
Dk 3x→z∈Sk
u(x), k = 1, m0 .
€
±
The notation T (∂z , n(z))u(z) has a similar meaning.
3. A Kupradze matrix of fundamental solutions of the homogeneous
equation of oscillation (2) has the form [8]


Γ(x − y, ω 2 ) = Γkj (x − y, ω 2 ) ,
3×3
where
Γkj (x − y, ω 2 ) =
ρδkj eik2 r
∂2
1
eik1 r − eik2 r
−
,
4πµ r
4πω 2 ∂xk ∂xj
r
(3)
i is the imaginary unit, r = |x − y|, k1 and k2 are the nonnegative numbers
defined by the equalities
k12 =
ρω 2
ρω 2
, k22 =
.
λ + 2µ
µ
(4)
Let κ0 be an arbitrary real fixed positive integer and κ > κ0 be an
arbitrary number. If in (3) we replace ω = iκ, then we obtain
κr
κr
κr
ρδkj e− c2
1
∂2
e− c1 − e− c2
+
,
Γkj (x − y, −κ ) =
4πµ r
4πκ 2 ∂xk ∂xj
r
2
(5)
where c21 = (λ + 2µ)ρ−1 , c22 = µρ−1 . Since µ > 0 and 3λ + 2µ > 0, we have
−1
1
λ + µ > 0 and λ + 2µ > µ. Hence c1 > c2 and c−1
1 < c2 . Let δ < 2c1 be
an arbitrary positive integer. Then (5) can be rewritten as
ρδkj −ακr e−δ1 κr
e
+
4πµ
r
 −δκr
1
∂2
e−δ1 κr ‘
−ακr e
+
,
e
−
4πκ 2 ∂xk ∂xj
r
r
Γkj (x − y, −κ 2 ) =
where δ1 = c12 − α =
Moreover,
1
c2
−
1
c1
+ δ > 0.
(6)
110
T. BURCHULADZE AND R. RUKHADZE
∂  −ακr e−δ1 κr ‘ e−ακr −δ1 κr 
1 ‘ ∂r
e
e
=
−
κr
;
−
1
∂xj
r
r2
c2
∂xj
3
∂ 2  −ακr e−δ1 κr ‘ e−ακr −δ1 κr 
3 + κr+
e
e
(2)
=
3
∂xk ∂xj
r
r
c2
‘

−ακr
∂r ∂r
1
e
∂r
1 ‘
−δ1 κr
1
+ 2 κ 2 r2
+
e
−
−
κr δkj
,
3
c2
∂xj ∂xk
r
c2
∂xk
(1)
(3) the functions (κr)n e−δ1 κr , n = 0, 1, 2, . . . are bounded in the interval κ ∈ [0, +∞).
Taking into account the above arguments, from (6) we obtain the estimates
Œ ∂ n Γ (x − y, −κ 2 ) Œ const
Œ
Œ
pq
(7)
Œ ≤ n+1 e−ακr ,
Œ
r
∂xi1 ∂xj2 ∂xk3
i + j + k = n;
n = 0, 1, 2, . . . ;
p, q = 1, 3.
4. Let x, y ∈ Dk , k = 0, m0 and ly be the distance from the point y to
the boundary of Dk .
Denote ρy (x) = max{r, ly }. and introduce an auxiliary matrix
”
•

bk
rm ‘n k
Γ(x − y, −κ 2 ) = 1 − 1 − m
Γ(x − y, −κ 2 ),
(8)
ρy (x)
k
where Γ(x − y, −κ 2 ) is the Kupradze matrix of fundamental solutions for
k
the operator A(∂x) − κ 2 I (I is the 3 × 3 unit matrix).
We denote by B(y, ly ) a sphere of radius ly and center €at the point
m n
y, and by C(y, ly ) its boundary. It is easy to verify that 1 − ρmr (x)
y
vanishes together with its derivatives up to the (n − 1)-th order inclusive
when the point x ∈ B(y, ly ) tends to the point of the boundary C(y, ly ).
For x ∈ Dk \B(y, ly ) we have

r m ‘n
1− 1− m
=1
ρy (x)
and
lim
B(y,ly )3x→z∈C(y,ly )
Thus
”

1− 1−
•
r m ‘n
= 1.
ρm
y (x)
bk
k
Γ(x − y, −κ 2 ) = Γ(x − y, −κ 2 )
for x ∈ Dk \B(y, ly ), while, in passing the boundary C(y, ly ), the function
bk
Γ and its derivatives up to the (n − 1)th order inclusive remain continuous.
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
111
bk
We write Γ as
bk
k
€

Γ(x − y, −κ 2 ) = Γ(x − y, −κ 2 ) nrm /ρm
y (x) + · · · .
bk
It is easy ti find that Γ and its derivatives up to the (m − 2)-th order
inclusive are continuous for x = y, while for x ∈ B(y, ly ) we have, by virtue
of (7), the estimates
bk
Œ ∂sΓ
2 Œ
const e−ακr m−s−1
Œ
pq (x − y, −κ ) Œ
,
≤
r
Œ
Œ
j
lym
∂xi1 ∂x2 ∂x3k
p, q = 1, 3,
i + j + k = s;
m ≥ s + 1.
5. We determine the limit
hk
i
k
lim Γ(x − y, −κ 2 ) − Γ(x − y, −κ02 ) , x, y ∈ Dk , k = 0, m0 .
x→y
Taking into consideration the expansion
κr
κ3 2
1
κ
κ2
e− c1
r−
r + ··· ,
= −
+
2
r
r
c1
2! c1
3! c31
we obtain
κ0 r
κr
−
e c2
κ0 − κ
κ 2 − κ02
e − c2
+
(1)
r + ···,
−
=
r
r
c2
2! c22
κr
κr
 1

1
1‘
1 ‘
e− c1
e− c2
−
=κ −
+ κ2
−
r+
(2)
+
r
r
c1
c2
2! c12
2! c22
‘

1
1
+κ 3 −
+
r2 + · · · ,
3
3! c1
3! c32
κr
κr
 1
1 ‘ ∂2r
∂2
e− c1 − e− c2
= κ2
−
(3)
+
2
∂xk ∂xj
r
2! c1
2! c22 ∂xk ∂xj
 1
1 ‘ ∂ 2 r2
−
+κ 3
+ ··· ,
6c23
6c13 ∂xk ∂xj
(4)
∂ 2 r2
= 2δkj .
∂xk ∂xj
By virtue of the above relations we have
hk
i
k
lim Γpq (x − y, −κ 2 ) − Γpq (x − y, −κ02 ) =
x→y
3/2
=
(κ0 − κ)ρk
3/2
4πµk
δpq +
3/2 ”
(κ0 − κ)ρk
12π
•
1
1
−
δpq =
3/2
(λk + 2µk )3/2
µk
(9)
112
T. BURCHULADZE AND R. RUKHADZE
=
3/2 ”
(κ0 − κ)ρk
12π
•
1
2
+
δpq ,
3/2
(λk + 2µk )3/2
µk
p, q = 1, 3.
(10)
6. Our further investigation will be carried out for the first problem.
The other problems are treated analogously.
We apply the term “Green’s tensor of the first basic boundary-contact
k
problem of the operator A(∂x) − κ02 I ” to the 3 × 3 matrix G(x, y, −κ02 ) =
k
€

m0
6 y, k ∈ 0, m0 which satisfies
G(x, y, −κ02 ), x ∈ Dk , y ∈ D D = ∪ Dk , x =
k=0
the following conditions:
(1) ∀x ∈ Dk , ∀y ∈ D, x 6= y :
k
k
k
A(∂x)G(x, y, −κ02 )−κ02 G(x, y, −κ02 ) = 0, k = 0, m0 ,
◦
k
(2) ∀z ∈ Sk , ∀y ∈ D : G+ (x, y, −κ02 ) = G − (x, y, −κ02 ),
◦
k
€◦
+ € k
−
T (∂z , n(z))G(z, y, −κ02 ) = T (∂z , n(z))G(z, y, −κ02 ) , k = 1, m0 ,
◦
(3) ∀z ∈ Sk , ∀y ∈ D : G+ (z, y, −κ02 ) = 0, k = 0, m0 + 1, . . . , m;
k
k
k
(4) G(x, y, −κ02 ) = Γ(x − y, −κ02 )− g(x, y, −κ02 ), x ∈ Dk , y ∈ D, k = 0, m0 ,
k
where g(x, y, −κ02 ) is a regular in Dk solution of the following problem:
k
k
k
(1) ∀x ∈ Dk , ∀y ∈ D : A(∂x)g(x, y, −κ02 )−κ02 g(x, y, −κ02 ) = 0, k = 0, m0 ;
◦
k
(2) ∀z ∈ Sk , ∀y ∈ D : g + (z, y, −κ02 ) − g − (z, y, −κ02 ) =
◦
k
= Γ(z − y, −κ02 ) − Γ(z − y, −κ02 ),
◦
+ € k
−
€
◦
k
T (∂z , n(z))g(z, y, −κ02 ) − T (∂z , n(z))g(z, y, −κ02 ) =
◦
◦
k
k
= T (∂z , n(z))Γ(z − y, −κ02 ) − T (∂z , n(z))Γ(z − y, −κ02 ), k = 1, m0 ;
◦
◦
(3) ∀z ∈ Sk , ∀y ∈ D : g + (z, y, −κ02 ) = Γ(z − y, −κ02 ),
k = 0, m0 + 1, . . . , m;
The solvability of this problem is shown in [8] and thereby the existence of
G(x, y, −κ02 ) is proved. As is known [8], G(x, y, −κ02 ) possesses a symmetry
property of the form
G(x, y, −κ02 ) = GT (y, x, −κ02 ).
(11)
Moreover, we have the estimates [9]
∀(x, y) ∈ Dk × Dk : Gpq (x, y, −κ02 ) = O(|x − y|−1 ),
∂
Gpq (x, y, −κ02 ) = O(|x − y|−2 ), m, n, j = 1, 3;
∂xj




k = 0, m0 .
(12)
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
k
113
k
7. Let u(x) = u(x) and v(x) = v(x), x ∈ Dk , be arbitrary (regular)
vectors of the class C 1 (Dk ) ∩ C 2 (Dk ), k = 0, m0 . Then the following Green
formula is valid [8]:
m0 Z
X
kkk
k k k
(v Au + E(v, u))dx =
k=0D
k
Z
◦
◦◦
v + (T u)+ ds +
S
m0 Z
X
€◦+ ◦ ◦ + k− k k −
+
v (T u) − v (T u) ds,
(13)
k=1S
k
where S = S0
m
∪
k=m0 +1
k k k
E(v, u) =
Sk ,
k
ρ−1 µk X  ∂ v p
ρk−1 (3λk + 2µk )
k
k
+
div v div u + k
3
2
∂xq
p6=q
k
k
X  ∂ vp
∂ uq ‘ ρ−1
∂ v q ‘ ∂ up
∂ uq ‘
∂ v q ‘ ∂ up
k µk
+
+
. (14)
+
−
−
∂xp ∂xq
∂xp
3
∂xp
∂xq
∂xp
∂xq
p,q
k
k
k
k
k
k k k
k k k
k k k
It follows from (14) that E(v, u) = E(u, v) and E(v, v) ≥ 0.
For the regular in Dk , k = 0, m0 vector u(x) the following general integral
representation is valid [8]:
∀y ∈ Dk : uj (y) = −
+
Z
S
m0 Z
X
k=0D
k
k
k
k
k
Γj (x − y, −κ 2 )(A(∂x)u(x) − κ 2 u(x))dx +
◦
‚k
◦
Γj (z − y, −κ 2 )(T (∂z , n(z))u(z))+ −
◦
◦
ƒ
◦
−u+ (z)T (∂z , n(z))Γj (z − y, −κ 2 ) dz S +
m0 Z
X
◦
‚◦
◦
+
Γj (z − y, −κ 2 )(T (∂z , n(z))u(z))+ −
k=1S
k
k
k
ƒ
k
−Γj (z − y, −κ 2 )(T (∂z , n(z))u(z))− dz S −
m0 Z
X
◦
◦
‚◦+
−
u (z)(T (∂z , n(z))Γj (z − y, −κ 2 ) −
k=1S
k
k
k
ƒ
k
−u− (z)(T (∂z , n(z))Γj (z − y, −κ 2 ) dz S, j = 1, 2, 3.
(15)
114
T. BURCHULADZE AND R. RUKHADZE
8. To establish the asymptotic behavior of eigenfunctions and eigenvalues
we have to estimate the regular part of Green’s tensor g(x, y, −κ 2 ) as κ →
∞. To this end we consider the functional
m0 Z
m0 Z
X
X
€k k k
‚◦+ ◦
k 
L[u] =
E(u, u) + κ 2 u2 dx − 2
u (z)T (∂z , n(z)) ×
k=0 D
k=1S
k
k
◦
k
k
ƒ
k
×Γj (z − y, −κ 2 ) − u− (z)(T (∂z , n(z))Γj (z − y, −κ 2 ) dz S,
(16)
where j = 1, 2, 3 is the fixed number and y is an arbitrary fixed point in Dk ,
k = 0, m0 , which is defined in the class of regular in Dk , k = 0, m0 , vector
functions satisfying the conditions:
k
◦
k
◦
(1) ∀z ∈ Sk : u+ (z) − u− (z) = Γj (z − y, −κ 2 ) − Γj (z − y, −κ 2 ),
kk
◦◦
◦
◦
(T u(z))+ − (T u(z))− = T (∂z , n(z))Γj (z − y, −κ 2 )−
k
k
T (∂z , n(z))Γj (z − y, −κ 2 ), k = 1, m0 ,
◦
◦
(2) ∀z ∈ Sk : u+ (z) = Γj (z − y, −κ 2 ), k = 0, m0 + 1, . . . , m.
Theorem 1. The functional L takes a minimal value for u = gj (x, y,−κ 2 ).
Proof. Let u be an arbitrary vector from the domain of definition of the
functional L, and let v = u − gj . Then, with (14) taken into account, (16)
implies
m0 Z
X
ƒ
€k k k k k
k
k
L[u] = L[v + gj ] =
E(v + g j , v + g j ) + κ 2 (v + g j )2 dx −
−2
=
m0 Z
X
k=1S
k
k=0D
k
‚
kk ƒ
k
k
◦
◦ ◦◦
(v + g j )T Γj − (v − + g j − )T Γj ds =
k k k
k k k
‚k k k
ƒ
k
kk
k
E(v, v) + 2E(v, g j ) + E(g j , g j ) + κ 2 (v 2 + 2v g j + g j 2 ) dx −
k=0D
k
m0 Z
X
−
m0 Z
X
k=1S
k
m0
X
‚◦+ ◦ ◦
k kk ƒ
v T Γj − v − T Γj ds − 2
= L[gj ] +
m0 Z
X
k=0D
k
Z
k=1S
k
kk ƒ
€◦ + ◦ ◦
k
g j T Γj − g j − T Γj ds =
m0
X
‚k k k
k ƒ
E(v, v) + κ 2 v 2 dx + 2
kk ƒ
+κ 2 v g j dx − 2
m0 Z
X
k=1S
k
Z
k=0D
k
‚k k k
E(v, g j ) +
‚◦+ ◦ ◦
k kk ƒ
v T Γj − v − T Γj ds.
(17)
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
115
Using the Green formula (13) for v = u − gj , u = gj and taking into
kk
k
◦
account that Ag j = κ 2 g j , v + (z) = 0 for z ∈ Sk , k = 0, m0 + 1, . . . , m, we
obtain
m0 Z
m0 Z
X
X
‚k k k
‚◦+ ◦ ◦ + k− k k −ƒ
kk ƒ
E(v, g j ) + κ 2 v g j dx =
v (T g j ) − v (T g j ) ds.
(18)
k=0D
k
k=1S
k
◦
◦◦
k
◦◦
kk
kk
Now, since v + (z) = v − (z), (T g j )+ − (T g j )− = T Γj − T Γj for z ∈ Sk ,
k = 1, m0 (17) by virtue of (18) takes the form
m0 Z
X
‚k k k
k ƒ
E(v, v) + κ 2 v 2 dx ≥ L[gj ].
L[u] = L[gj ] +
k=0D
k
Theorem 2. The estimate
Œ
Œ
Œgjj (y, y, −κ 2 ) − gjj (y, y, −κ02 )Œ ≤ const , y ∈ D,
ly1+δ
δ > 0,
(19)
holds for the function gjj (y, y, −κ 2 ).
Proof. We write formula (15) for uj (x) = gjj (x, y, −κ 2 ) and Γj (x−y, −κ 2 ) =
Gj (x − y, −κ 2 ). Then taking into account the boundary and contact conditions for g and G, we get
=−
Z
◦
∀(x, y) ∈ Dk : gjj (x, y, −κ 2 ) =
◦
◦
Γj (z − x, −κ 2 )(T (∂z , n(z))Gj (z, y, −κ 2 ))+ dz S −
S
m0 Z
X
◦
◦
‚◦ +
Gj (z, x, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) −
+
k=1S
k
k
k
k
ƒ
−Gj − (z, x, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) dz S −
m0 Z
X
◦
◦
‚◦
−
Γj (z − x, −κ 2 )(T (∂z , n(z))Gj (z, y, −κ 2 ))+ −
k=1S
k
k
k
k
ƒ
−Γj (z − x, −κ 2 )(T (∂z , n(z))Gj (z, y, −κ 2 ))− dz S.
(20)
Using formula (13) for u = v, (16) can be rewritten as
Z
m0 Z
X
◦◦
ƒ
k‚ k
◦
k
k
u A(∂x )u(x) − κ 2 u(x) dx + u+ (z)(T u(z))+ dz S +
L[u] = −
k=0D
k
S
116
T. BURCHULADZE AND R. RUKHADZE
+
m0 Z
X
kk
◦◦
ƒ
‚◦+
k
u (z)(T u(z))+ − u− (z)(T u(z))− dz S −
k=1S
k
m0 Z
X
−2
k=1S
k
◦
‚◦+ ◦
u (z)T (∂z , n(z))Γj (z − y, −κ 2 ) −
k
k
ƒ
k
−u− (z)T (∂z , n(z))Γj (z − y, −κ 2 ) dz S.
(21)
which for u(x) = gj (x, y, −κ 2 ) = Γj (x − y, −κ 2 ) − Gj (x, y, −κ 2 ) implies
Z ◦
◦
◦
L[gj ] = Γj (z − y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 )dz S −
−
Z
S
◦
+
€◦
Γj (z − y, −κ 2 ) T (∂z , n(z))Gj (z, y, −κ 2 ) dz S +
◦
S
m0 Z
X
+
k=1S
k
◦
◦
‚◦
Γj (z − y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) −
k
k
k
ƒ
−Γj (z − y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) dz S +
m0 Z
X
◦
◦
‚◦ +
Gj (z, y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) −
+
k=1S
k
k
k
k
ƒ
−Gj − (z, y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) dz S −
m0 Z
X
◦
◦
‚◦
Γj (z − y, −κ 2 )(T (∂z , n(z))Gj (z, y, −κ 2 ))+ −
−
k=1S
k
k
k
k
ƒ
−Γj (z − y, −κ 2 )(T (∂z , n(z))Gj (z, y, −κ 2 ))− dz S.
(22)
On the basis of (22), from (20) we obtain
Z ◦
◦
◦
2
gjj (y, y, −κ ) = L[gj ] − Γj (z − y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 )dz S +
S
m0 Z
X
◦
◦
‚◦
+
Γj (z − y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) −
k=1S
k
k
k
k
ƒ
−Γj (z − y, −κ 2 )T (∂z , n(z))Γj (z − y, −κ 2 ) dz S.
(23)
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
117
bk
The vector Γj (x − y, −κ 2 ) defined by (8) belongs to the domain of definition of the functional L and, since gj (x, y, −κ 2 ) imparts a minimal value
to the functional L, it is obvious that
b j ].
L[gj ] ≤ L[Γ
Now (23) implies
bj ] −
gjj (y, y, −κ ) ≤ L[Γ
2
+
m0 Z
X
◦ ◦◦
(Γj T Γj
k=1S
k
−
Z
◦ ◦◦
Γj T Γj ds +
S
k kk
Γj T Γj )ds,
y ∈ Dk .
(24)
b from (21) we obtain
By virtue of the properties of Γ,
Z
Z ◦ ◦◦
b j )dx + Γj T Γj ds −
b j (AΓ
bj − κ2 Γ
bj ] = −
Γ
L[Γ
S
B(y,ly )
−
m0 Z
X
k=1S
k
◦ ◦◦
k kk
(Γj T Γj − Γj T Γj )ds.
(25)
By (25) and (24) we have
Z
b j )dx, y ∈ Dk , k = 0, m0 . (26)
b j (AΓ
bj − κ2 Γ
gjj (y, y, −κ 2 ) ≤ −
Γ
B(y,ly )
Taking into account estimates (9), for m = 5 we obtain
Œ
Πconst
Œb
Γmj (x, y, −κ 2 )Œ ≤
for s = 0,
ly
−ακr
Π2
Œ
b ij (x, y, −κ 2 )Œ ≤ κ 2 const e
Œκ Γ
r4 =
ly5
const
const
= 5 r2 (κ 2 r2 )e−ακr ≤ 3
for s = 0,
ly
ly
Œ
Œ
b j (x, y, −κ 2 )Œ ≤ const for s = 2.
ŒAΓ
ly3
Hence (26) implies
gjj (y, y, −κ 2 ) ≤
const
const 4 3
const
· πly ≤
≤ 1+δ ,
ly4
3
ly
ly
where δ > 0 is an arbitrary integer.
(27)
118
T. BURCHULADZE AND R. RUKHADZE
Let us estimate gjj (y, y, −κ 2 ) from below. For this we introduce the
notation:
m0 Z
m0 Z
X
X
‚k k k
‚k k k
k ƒ
k ƒ
M [u] =
E(u, u) + κ 2 u2 dx, M0 [u] =
E(u, u) + κ02 u2 dx,
k=0D
k
m0 Z
X
N [u] =
k=1S
k
k=0D
k
kk
‚◦+ ◦ ◦
ƒ
k
u (z)T Γj (z − y, −κ 2 ) − u− (z)T Γj (z − y, −κ 2 ) dz S.
Since κ02 ≤ κ 2 , we have
L[gj (x, y, −κ 2 )] = min L[u] = min(M [u] − 2N [u]) ≥ min(Mo [u] − 2N [u]).
Let the vector function ϕ(x, y) impart a minimal value to the functional
M0 [u] − 2N [u]. Then ϕ(x, y) is a regular in Dk (k = 0, m0 ) solution of the
following problem:
k
k
k
(1) ∀x ∈ Dk , ∀y ∈ D : A(∂x)ϕ(x, y) − κ02 ϕ(x, y) = 0, k = 0, m0 ;
◦
◦
k
k
(2) ∀z ∈ Sk , ∀y ∈ D : ϕ+ (z, y)− ϕ− (z, y) = Γ(z −y, −κ 2 )− Γ(z −y, −κ 2 ),
◦◦
kk
◦◦
kk
(T ϕ(z, y))+ − (T ϕ(z, y))− = T Γ(z − y, −κ 2 ) − T Γ(z − y, −κ 2 );
◦
◦
(3) ∀z ∈ Sk , ∀y ∈ D : ϕ+ (z, y) = Γ(z − y, −κ 2 ), k = 0, m0 + 1, . . . , m.
Rewriting formula (15) for ϕ(x, y) with Γ = G, we obtain
Z ◦
◦ ◦
∀(x, y) ∈ Dk : ϕ(x, y) = − Γj (z − x, −κ 2 )T Gj (z, y, −κ02 )dz s +
S
m0 Z
X
k
◦◦
‚◦
Gj (z, x, −κ02 )T Γj (z, y, −κ 2 ) − Gj − (z, x, −κ02 ) ×
+
k=1S
k
m0
X
kk
ƒ
×T Γj (z − y, −κ 2 ) dz s −
Z
k=1S
k
◦ ◦
‚◦
Γj (z − x, −κ 2 )(T Gj (z, y, −κ02 ))+ −
k
k k
ƒ
−Γj (z − x, −κ 2 )(T Gj (z, y, −κ02 )) dz s.
(28)
By (7) and (12) and the theorem on kernel composition [10] it follows
from (28) that
∀(x, y) ∈ Dk × Dk : |ϕ(x, y)| ≤
const
, k = 0, m0 , rxy = |x − y|. (29)
rxy
In that case
L[gj (x, y, −κ 2 ) ≥ M0 [ϕ] − 2N [ϕ] ≥ −2N [ϕ] =
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
= −2
119
m0 Z
X
kk
◦◦
‚◦+
ƒ
k
ϕ (z, y)T Γj (z −y,−κ 2 ) − ϕ− (z, y)T Γj (z −y,−κ 2 ) dz s. (30)
k=1S
k
Now taking into account that
const
1
≤ ,
rzy
ly
const
const
∀(z, y) ∈ Sk × Dk : |T Γj (z − y, −κ 2 )| ≤ 2 =
≤
2−δ
δ
rzy
rzy rzy
∀(z, y) ∈ Sk × Dk : |ϕ(z, y)| ≤
≤
1
1
· 2−δ , δ > 0,
δ
ly rzy
from (30) we obtain
L[gj ] ≥ −
const
,
ly1+δ
δ > 0.
By virtue of representation (23) we can easily conclude that the estimate
∀y ∈ D : gjj (y, y, −κ 2 ) ≥ −
const
ly1+δ
(31)
holds.
(27) and (31) imply (19).
9. Consider the first boundary-contact problem on eigenvalues: In the
k
domain Dk (k = 0, m0 ), find a regular vector w(x) = w(x), x ∈ Dk , which
is a nontrivial solution of the equations
k
k
k
∀x ∈ Dk : A(∂x)w(x) + γ w(x) = 0,
k = 0, m0 ,
satisfying the contact conditions
◦
k
∀z ∈ Sk : w+ (z) = w− (z),
◦ ◦
k k
(T w(z))+ = (T w(z))− , k = 1, m0 ,
and the boundary condition
◦
∀z ∈ Sk : w+ (z) = 0, k = 0, m0 + 1, . . . , m.
We denote this problem by Iγ . In the same manner as in [8] we can show
that problem Iγ is equivalent to a system of integral equations
Z
2
(32)
w(x) = (γ + κ0 ) G(x, y, −κ02 )w(y)dy.
D
120
T. BURCHULADZE AND R. RUKHADZE
By virtue of (11) and (12) equation (32) is an integral equation with
a symmetric kernel of the class L2 (D). Hence it follows that there exists a countable system of eigenvalues (γn + κ02 )∞
n=1 and the corresponding
k
(n)
orthonormal in D system of eigenvectors {w(n) (x)}∞
(x)}∞
n=1 = {w
n=1 ,
(n)
∞
x ∈ Dk , k = 0, m0 , of equation (32). Therefore (γn )n=1 and {w (x)}∞
n=1
are respectively the eigenvalues and eigenvectors of problem Iγ . As established in [8], all γn > 0. Moreover, it is proved in [11] that the system
{w(n) (x)}∞
n=1 is complete in L2 (D). The properties of the volume potential
[8] imply that eigenvectors are regular.
10. Hardy and Littlewood’s theorem of the Tauber type [12] plays the
major part in deriving asymptotic formulas.
Theorem 3. If a nondecreasing function Φ(t) is Stieltjes summable, and
for x → ∞ the asymptotic representation
Z∞
0
dΦ(t)
p
∼ m,
l
(x + t)
x
where l, m, p satisfy the conditions 0 < m < l, p =
6 0, is fulfilled, then
Φ(t) ∼
Here Γ is the Euler function.
pΓ(l)
tl−m .
Γ(m)Γ(l − m + 1)
Write the kernel expansion in terms of the eigenfunctions
G(x, y, −κ 2 ) =
G(x, y, −κ02 ) =
∞
X
w(n) (x) × w(n) (y)
,
γn + κ 2
n=1
∞
X
w(n) (x) × w(n) (y)
,
γn + κ02
n=1
(33)
(34)
where x, y ∈ Dk , k = 0, m0 , and the symbol × denotes the matrix product
of a column vector by a row vector (dyadic product) w(n) (x) × w(n) (y) =
(n)
(n)
kwi (x) · wj (y)k i, j = 1, 2, 3. From (33) and (34) we obtain
G(x, y, −κ 2 ) − G(x, y, −κ02 ) = (κ02 − κ 2 )
∞
X
w(n) (x) × w(n) (y)
. (35)
(γn + κ 2 )(γn + κ02 )
n=1
Passage in equality (35) to the limit as x → y results in
∞
X
(n)
‚
[wj (y)]2
= lim Γjj (x − y, −κ 2 ) −
2
2
(γn + κ )(γn + κ0 ) x→y
n=1
ƒ
‚
ƒ
−Γjj (x − y, −κ02 ) − lim gjj (x, y, −κ 2 ) − gjj (x, y, −κ02 ) ,
(κ02 − κ 2 )
x→y
(36)
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
121
x, y ∈ Dk , k = 0, m0 , j = 1, 3.
Taking into account (10) and (19), we find from (36) that
”
•
(n)
∞
3/2
X
[wj (y)]2
ρk
2
1
∼
+
, (37)
3/2
(γn + κ 2 )(γn + κ02 )
12π(κ + κ0 ) (λk + 2µk )3/2
µk
n=1
y ∈ Dk , k = 0, m0 , j = 1, 3.
Consider the function
X [wj(n) (y)]2
γn + κ02
Φj (t) =
γn ≤t
y ∈ Dk , k = 0, m0 , j = 1, 3.
It is easy to observe that
Z∞
(n)
∞
[wj (y)]2
dΦj (t) X
=
.
t + κ2
(γn + κ 2 )(γn + κ02 )
n=1
0
By (37) we have
Z∞
0
dΦj (t)
Ak
∼
,
t + κ2
κ
(38)
where
•
3/2 ”
ρk
1
2
Ak =
+ 3/2 .
12π (λk + 2µk )3/2
µk
By Theorem 3 with regard for (38) we obtain
X [wj(n) (y)]2
2
Ak Γ(1) 1/2
Φj (t) =
∼
= Ak t1/2 .
3 t
1
γn + κ02
π
Γ( 2 )Γ( 3 )
γ ≤t
(39)
n
Obviously,
X
(n)
[wj (y)]2
γn ≤t
=
Zt
(ξ +
0
κ02 )dΦj (ξ)
= (ξ +
Œt
κ02 )Φj (ξ)Œ0
−
Zt
Φj (ξ)dξ.
0
Taking into account (39), we obtain
X (n)
2
Ak t3/2 , y ∈ Dk , k = 0, m0 , j = 1, 3.
[wj (y)]2 ∼
3π
γn ≤t
If we sum these relations with respect to j = 1, 2, 3, then we shall have
X (n)
2
[wj (y)]2 ∼ Ak t3/2 , y ∈ D,
π
γn ≤t
122
T. BURCHULADZE AND R. RUKHADZE
or
X
k
[wj (n) (y)]2
γn≤t
•
3/2 ”
ρk
2
1
+
t3/2 , y ∈ Dk , k = 0, m0 . (40)
∼
6π 2 (λk +2µk )3/2 µ3/2
k
The above expression provides the asymptotic distribution of eigenfunctions.
11. Taking into account (10), from (36) we obtain
(κ 2 − κ02 )
∞
X
[w(n) (y)]2
= 3(κ − κ0 )Ak +
(γn + κ02 )(γn + κ 2 )
n=1
3
X
‚
ƒ
gjj (y, y, −κ02 ) − gjj (y, y, −κ 2 ) .
+
(41)
j=1
Denote
ϕ(y, κ) = 3(κ − κ0 )Ak +
3
X
‚
ƒ
gjj (y, y, −κ02 ) − gjj (y, y, −κ 2 ) .
j=1
Now (41) yields
∞
∞
X
X
ϕ(y, κ)
[w(n) (y)]2
[w(n) (y)]2
≤
.
=
κ 2 − κ02
(γn + κ02 )(γn + κ 2 ) n=1 (γn + κ02 )2
n=1
By virtue of the Bessel inequality we have
Z
∞
X
[w(n) (x)]2
≤
|G(x, y, −κ02 )|2 dy,
2 )2
(γ
+
κ
n
0
n=1
x ∈ Dk , k = 0, m0 .
(42)
(43)
D
By estimate (8) it follows from (42) that the sum of the series
∞
X
[w(n) (x)]2
(γn + κ02 )2
n=1
exists and is uniformly bounded in Dk . This and (42) imply
∀y ∈ Dk : |ϕ(y, κ)| ≤ const(κ 2 − κ02 ),
k = 0, m0 .
(44)
Integrating equality (42) in D and using the fact that the vectors
[w(n) (x)]∞
n=1 are orthonormal in D, we obtain
Z
∞
X
1
,
ϕ(y, κ)dy = (κ 2 − κ02 )
(45)
2
(γn + κ0 )(γn + κ 2 )
n=1
D
Z
D
ϕ(y, κ)dy =
Z
D
Z X
3
‚
gjj (y, y, −κ02 ) −
3(κ − κ0 )Ak dy +
D j=1
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
123
m0
X
ƒ
−gjj (y, y, −κ 2 ) dy = 3(κ − κ0 )
Ak mes Dk +
k=0
+
3 Z
X
j=1 D
‚
ƒ
gjj (y, y, −κ02 ) − gjj (y, y, −κ 2 ) dy.
(46)
By (46), from (45) we have
m0
X
3
1
Ak mes Dk =
−
(γn + κ02 )(γn + κ 2 ) κ + κ0
n=1
k=0
3 Z
X
‚
ƒ
1
= 2
gjj (y, y, −κ02 ) − gjj (y, y, −κ 2 ) dy.
2
κ − κ0 j=1
∞
X
(47)
D
Denote by Dkη that part of Dk , k = 0, m0 , whose points lie from the
m0
boundary Dk at a distance less than η. Dη = ∪ Dkη . Then
k=0
Z
D
ƒ
‚
gjj (y, y, −κ02 ) − gjj (y, y, −κ 2 ) dy =
ƒ
−gjj (y, y, −κ 2 ) dy +
Z
Z
D\Dη
ϕ(y, κ)dy −
Z
‚
gjj (y, y, −κ02 ) −
3(κ − κ0 )Ak dy.
Dη
Dη
This and (47) imply
Œ
Œ ∞
m0
X
Œ
ŒX
3
1
Œ
Œ
mes
D
−
A
kŒ ≤
k
2
Œ
2
(γn + κ0 )(γn + κ ) κ + κ0
n=1
k=0
ΠZ
Œ
ŒZ
3
XŒ
Œ
Œ
‚
1
1
Œ
Œ ϕ(y, κ)dy Œ +
≤ 2
gjj (y, y, −κ02 ) −
2
2
Œ
Œ
Œ
2
κ − κ0
κ − κ0 j=1
Dη
Œ
ƒ Œ
−gjj (y, y, −κ) dy ŒŒ +
D\Dη
ŒZ
Œ
Œ
3 ŒŒ
Œ
A
dy
k
Œ
Œ.
κ + κ0
(48)
Dm
The following estimates are valid:
ŒZ
Πm0 Z
Œ
Œ
Œ
Œ
3 ŒŒ
3 ŒŒ X
3
Œ
Ak dy Π=
η,
Ak dy ŒŒ ≤ const
Œ
Œ
κ + κ0
κ + κ0
κ + κ0
(49)
k=0D
kη
Dη
ŒZ
Œ
Œ
Œ
1
Œ ϕ(y, κ)dy Œ ≤ const (κ 2 − κ02 )η = const η,
2
Œ κ2 − κ2
Œ
2
κ − κ0
0
Dη
(50)
124
T. BURCHULADZE AND R. RUKHADZE
3 ΠZ
X
Œ
1
Œ
2
Œ
2
κ −κ0 j=1
D\Dη
‚
Œ
ƒ Œ
const 1
dy ŒŒ ≤ 2
. (51)
κ −κ02 η δ
gjj (y, y,−κ02 )−gjj (y, y,−κ)
The validity of (49) is obvious. (50) and (51) hold by virtue of (44)
and (19), respectively. It is important to remark here that the constants
appearing in (49), (50) and (51) do not depend on κ and y. Consider the
function
X
1
Φ(t) =
.
γn + κ02
γn ≤t
It is easy to verify that
Z∞
0
∞
X
dΦ(t)
1
=
2 )(γ + κ 2 ) .
+
κ
t + κ2
(γ
n
n
0
n=1
By of (49), (50) and (51) we find from (48) that
Z∞
0
for η =
1
.
κ 2 −κ02
3
dΦ(t)
∼
t + κ2
Pm0
Ak mes Dk
κ
k=0
According to Theorem 3 we have
Φ(t) ∼
Denoting
6
Pm0
k=0
N (t) =
Ak mes Dk 1/2
t .
π
X
(52)
1,
γn ≤t
for a number of eigenvalues not higher than t, we get
N (t) =
Zt
(ξ +
κ02 )dΦ(ξ)
= (ξ +
0
Hence with regard for (52) we obtain
N (t) ∼
Œt
κ02 )Φ(ξ)Œ0
−
Zt
Φ(ξ)dξ.
0
m0
2X
Ak mes Dk · t3/2 ,
π
k=0
or, finally,
•
m0 ”
1 X
2
1
3/2
+ 3/2 ρk mes Dk · t3/2 .
N (t) ∼
3/2
6π 2
+
2µ
)
(λ
k
k
µk
k=0
Thus the results of this paper can be formulated as
(53)
ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS
125
Theorem 4. The asymptotic distribution of eigenvector-functions and
eigenvalues of the basic boundary-contact problems of oscillation of classical
elasticity is given by formulas (40) and (53), respectively.
Remark 1. The results obtained remain valid (a) when the boundary
conditions on Sk (k = 0, m0 + 1, . . . , m) are replaced by those of the second
basic problem [8] (i.e., with elastic stresses given on the boundary) and
(b) when the conditions of the first problem are given on some part of
the boundary Sk (k = 0, m0 + 1, . . . , m), and the conditions of the second
problem on the remainder of the boundary (a mixed problem).
Thus we have considered the case of the main contact conditions. The
results remain valid for other admissible contact conditions [13].
Remark 2. If λk = λ, µk = µ, ρk = ρ (k = 0, m0 ) i.e., if a homogeneous
m0
m0
elastic medium D0 ∪ ( ∪ Dk ) ∪ ( ∪ Sk ) is considered, then, as can readily
k=1
[w(n) (x)]∞
n=1 and
k=1
(γn )∞
n=1
will be the eigenvector-functions and
be verified,
eigenvalues of equation (2) the corresponding boundary value problem and
the obtained asymptotic formulas will coincide with the well-known formulas
of Weyl, Plejel, Burchuladze and others.
Remark 3. Note that the above asymptotic formulas with certain modifications are also fulfilled for two-dimensional problems. For example, in
the two-dimensional case formula (53) takes the form
•
m0 ”
1
1 X
1
+
ρk mes Dk · t,
N (t) ∼
4π
λk + 2µk
µk
k=0
where mes Dk is a space of the domain Dk .
References
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126
T. BURCHULADZE AND R. RUKHADZE
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(Received 15.05.1997)
Authors’ addresses:
Tengiz Burchuladze
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, M. Aleksidze St., Tbilisi 380093
Georgia
Roland Rukhadze
Georgian Technical University
Department of Mathematics (3)
77, M. Kostava St., Tbilisi 380075
Georgia