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.

107

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.

1072-947X/99/0300-0107$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

2. Throughout the paper we shall use the following notation: x = (x_{1}, x_{2}, x_{3}), y = (y_{1}, y_{2}, y_{3}) are points of ℝ^{3}; |x − y| = ( ∑_{k=1}^{3}(x_{k} − y_{k})^{2})^{1/2} is the distance between the points x and y; D_{0} ⊂ ℝ^{3} is a finite domain bounded by closed surfaces S_{0}, S_{1}, …, S_{m} of the class Ω_{2}(α) 0 < α ≤ 1 [8] with S_{0} covering all other S_{k} while the latter surfaces not covering each other; S_{i} ∩ S_{k} = ∅ for i ≠ k, i, k = \overline{0, m}; the finite domain bounded by S_{k}

(k = \overline{1, m}) will be denoted by D_{k}; \overline{D}_{0} = D_{0} ∪ ( ∪_{k=0}^{m} S_{k}), \overline{D}_{k} = D_{k} ∪ S_{k},

k = \overline{1, m}.

If u and v are the three-component real vectors u = (u_{1}, u_{2}, u_{3}) and v = (v_{1}, v_{2}, v_{3}), then uv = ∑_{i=1}^{3} u_{i}v_{i} is the scalar product of these vectors; |u| = ( ∑_{i=1}^{3} u_{i}^{2})^{1/2}. The matrix product is obtained by multipliing a row by a column; the sign [·]^{T} denotes the operation of transposition; if A = ∥A_{ij}∥_{3×3} is a 3 × 3 matrix, then |A|^{2} = ∑_{i,j=1}^{3} A_{ij}^{2}. Any vector u = (u_{1}, u_{2}, u_{3}) is treated as a 3 × 1 one-column matrix: u = ∥u_{i}∥_{3×1}; A_{k} = ∥A_{jk}∥_{j=1}^{3} is the k-th column vector of the matrix A.

The vector u = (u_{1}, u_{2}, u_{3}) will be called regular in D_{k} if

u_{i} ∈ C^{1}(\overline{D}_{k}) ∩ C^{2}(D_{k}), 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]

μΔu + (λ + μ) grad div u + ρω^{2}u = 0,  (1)

where u(x) = (u_{1}, u_{2}, u_{3}) is the displacement vector, Δ is the three-dimen-sional 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) = ∥A_{ij}(∂x)∥_{3×3}, A_{ij}(∂_{x}) = δ_{ij}μρ^{−1}Δ + (λ + μ)ρ^{−1} \frac{∂^{2}}{∂x_{i} ∂x_{j}} ,

where δ_{ij} is the Kronecker symbol. Then equation (1) can be rewritten in the vector-matrix form

A(∂_{x})u + ω^{2}u = 0.  (2)

The matrix-differential operator

T (∂_{x}, n(x)) = ∥T_{ij}(∂_{x}, n(x))∥_{3×3},

where

T_{ij}(∂_{x}, n(x)) = λρ^{−1}n_{i}(x)\frac{∂}{∂x_{j}} + λρ^{−1}n_{j}(x)\frac{∂}{∂x_{i}} + μρ^{−1}δ_{ij} \frac{∂}{∂n(x)},

n(x) is an arbitrary unit vector at the point x (if x ∈ S_{k}, then k = \overline{0, m} is the normal unit vector external with respect to the domain D_{0}) is called the stress operator.

It will be assumed that the domains D_{k}, k = \overline{0, m_{0}} are filled with homogeneous isotropic elastic media with the Lamé constants λ_{k}, μ_{k} and density ρ_{k}, while the other domains D_{k}, k = \overline{m_{0} + 1, m} are hollow inclusions. When the operators A and T contain λ_{k} and μ_{k} instead of λ and μ, we shall write

A^{k} and T^{k}, respectively.

We introduce the notation

u^{+}(z) = lim_{D_{0}∋x→z∈S_{k}} u(x), k = \overline{0, m}; u^{−}(z) = lim_{D_{k}∋x→z∈S_{k}} u(x), k = \overline{1, m_{0}}.

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}) = ∥Γ_{kj}(x − y, ω^{2})∥_{3×3},

where

Γ_{kj}(x − y, ω^{2}) = \frac{ρδ_{kj}}{4πμ} \frac{e^{ik_{2}r}}{r} − \frac{1}{4πω^{2}} \frac{∂^{2}}{∂x_{k} ∂x_{j}} \frac{e^{ik_{1}r} − e^{ik_{2}r}}{r} ,  (3)

i is the imaginary unit, r = |x − y|, k_{1} and k_{2} are the nonnegative numbers defined by the equalities

k_{1}^{2} = \frac{ρω^{2}}{λ + 2μ}, k_{2}^{2} = \frac{ρω^{2}}{μ} .  (4)

Let κ_{0} be an arbitrary real fixed positive integer and κ > κ_{0} be an arbitrary number. If in (3) we replace ω = iκ, then we obtain

Γ_{kj}(x − y, −κ^{2}) = \frac{ρδ_{kj}}{4πμ} \frac{e^{− \frac{κr}{c_{2}}}}{r} + \frac{1}{4πκ^{2}} \frac{∂^{2}}{∂x_{k} ∂x_{j}} \frac{e^{− \frac{κr}{c_{1}}} − e^{− \frac{κr}{c_{2}}}}{r} , (5)

where c_{1}^{2} = (λ + 2μ)ρ^{−1}, c_{2}^{2} = μρ^{−1}. Since μ > 0 and 3λ + 2μ > 0, we have λ + μ > 0 and λ + 2μ > μ. Hence c_{1} > c_{2} and c_{1}^{−1} < c_{2}^{−1}. Let δ < \frac{1}{2c_{1}} be an arbitrary positive integer. Then (5) can be rewritten as

Γ_{kj}(x − y, −κ^{2}) = \frac{ρδ_{kj}}{4πμ} e^{−ακr} \frac{e^{−δ_{1}κr}}{r} +

+\frac{1}{4πκ^{2}} \frac{∂^{2}}{∂x_{k} ∂x_{j}} e^{−ακr}(\frac{e^{−δκr}}{r} − \frac{e^{−δ_{1}κr}}{r} ),  (6)

where δ_{1} = \frac{1}{c_{2}} − α = \frac{1}{c_{2}} − \frac{1}{c_{1}} + δ > 0.

Moreover,

(1) \frac{∂}{∂x_{j}} (e^{−ακr} \frac{e^{−δ_{1}κr}}{r} ) = \frac{e^{−ακr}}{r^{2}} e^{−δ_{1}κr}( − 1 − \frac{1}{c_{2}} κr)\frac{∂r}{∂x_{j}};

(2) \frac{∂^{2}}{∂x_{k} ∂x_{j}} (e^{−ακr} \frac{e−δ^{1}κr}{r} ) = \frac{e^{−ακr}}{r^{3}} e^{−δ_{1}κr}(3 + \frac{3}{c_{2}} κr+

+\frac{1}{c_{2}^{2}} κ^{2}r^{2})\frac{∂r}{∂x_{j}} \frac{∂r}{∂x_{k}} + \frac{e^{−ακr}}{r^{3}} e^{−δ_{1}κr}( − 1 − \frac{1}{c_{2}} κr)δ_{kj} \frac{∂r}{∂x_{k}},

(3) the functions (κr)^{n}e^{−δ_{1}κr}, n = 0, 1, 2, … are bounded in the interval κ ∈ [0, +∞).

Taking into account the above arguments, from (6) we obtain the estimates

|\frac{∂^{n}Γ_{pq}(x − y, −κ^{2})}{∂x_{1}^{i} ∂x_{2}^{j} ∂x_{3}^{k}} | ≤ \frac{const}{r^{n+1}}e^{−ακr},  (7)
i + j + k = n; n = 0, 1, 2, …; p, q = \overline{1, 3}.

4. Let x, y ∈ D_{k}, k = \overline{0, m_{0}} and l_{y} be the distance from the point y to the boundary of D_{k}.

Denote ρ_{y}(x) = max{r, l_{y}}. and introduce an auxiliary matrix

_{k} ; Γ(x − y, −κ^{2}) = [1 − (1 − \frac{r^{m}}{ρ_{y}^{m}(x)})^{n}]Γ^{k}(x − y, −κ^{2}),  (8)

where Γ^{k}(x − y, −κ^{2}) is the Kupradze matrix of fundamental solutions for

the operator A^{k}(∂x) − κ^{2}I (I is the 3 × 3 unit matrix).

We denote by B(y, l_{y}) a sphere of radius l_{y} and center at the point y, and by C(y, l_{y}) its boundary. It is easy to verify that (1 − \frac{r^{m}}{ρ_{y}^{m}(x)})^{n} vanishes together with its derivatives up to the (n − 1)-th order inclusive when the point x ∈ B(y, l_{y}) tends to the point of the boundary C(y, l_{y}). For x ∈ D_{k}\\overline{B(y, l_{y})} we have

1 − (1 − \frac{r^{m}}{ρ_{y}^{m}(x)})^{n} = 1

and

lim_{B(y,l_{y})∋x→z∈C(y,l_{y})} [1 − (1 − \frac{r^{m}}{ρ_{y}^{m}(x)})^{n}] = 1.

Thus

_{k} ; Γ(x − y, −κ^{2}) = Γ^{k}(x − y, −κ^{2})

for x ∈ D_{k}\B(y, l_{y}), while, in passing the boundary C(y, l_{y}), the function

_{k} ; Γ and its derivatives up to the (n − 1)th order inclusive remain continuous.

We write _{k} ; Γ as

_{k} ; Γ(x − y, −κ^{2}) = Γ^{k}(x − y, −κ^{2})(nr^{m}/ρ_{y}^{m}(x) + ⋯).

It is easy ti find that _{k} ; Γ and its derivatives up to the (m − 2)-th order inclusive are continuous for x = y, while for x ∈ B(y, l_{y}) we have, by virtue of (7), the estimates

|\frac{∂^{s}_{k} ; Γ_{pq}(x − y, −κ^{2})}{∂x_{1}^{i} ∂x_{2}^{j} ∂x_{3}^{k}} | ≤ \frac{const e^{−ακr}}{l_{y}^{m}} r^{m−s−1},  (9)
p, q = \overline{1, 3}, i + j + k = s; m ≥ s + 1.

5. We determine the limit

lim_{x→y} [Γ^{k}(x − y, −κ^{2}) − Γ^{k}(x − y, −κ_{0}^{2})], x, y ∈ D_{k}, k = \overline{0, m_{0}}.

Taking into consideration the expansion

\frac{e^{− \frac{κr}{c_{1}}}}{r} = \frac{1}{r} − \frac{κ}{c_{1}} + \frac{κ^{2}}{2! c_{1}^{2}} r − \frac{κ^{3}}{3! c_{1}^{3}} r^{2} + ⋯,

we obtain

(1) \frac{e^{− \frac{κr}{c_{2}}}}{r} − \frac{e^{− \frac{κ_{0}r}{c_{2}}}}{r} = \frac{κ_{0} − κ}{c_{2}} + \frac{κ^{2} − κ_{0}^{2}}{2! c_{2}^{2}} r + ⋯,

(2) \frac{e^{− \frac{κr}{c_{1}}}}{r} − \frac{e^{− \frac{κr}{c_{2}}}}{r} = κ( − \frac{1}{c_{1}} + \frac{1}{c_{2}} ) + κ^{2}(\frac{1}{2! c_{1}^{2}} − \frac{1}{2! c_{2}^{2}} )r+

+κ^{3}( − \frac{1}{3! c_{1}^{3}} + \frac{1}{3! c_{2}^{3}} )r^{2} + ⋯,

(3) \frac{∂^{2}}{∂x_{k} ∂x_{j}} \frac{e^{− \frac{κr}{c_{1}}} − e^{− \frac{κr}{c_{2}}}}{r} = κ^{2}(\frac{1}{2! c_{1}^{2}} − \frac{1}{2! c_{2}^{2}} )\frac{∂^{2}r}{∂x_{k} ∂x_{j}} +

+κ^{3}(\frac{1}{6c_{2}^{3}} − \frac{1}{6c_{1}^{3}} )\frac{∂^{2}r^{2}}{∂x_{k} ∂x_{j}} + ⋯,

(4) \frac{∂^{2}r^{2}}{∂x_{k} ∂x_{j}} = 2δ_{kj}.

By virtue of the above relations we have

lim_{x→y} [Γ^{k}_{pq}(x − y, −κ^{2}) − Γ^{k}_{pq}(x − y, −κ_{0}^{2})] =

= \frac{(κ_{0} − κ)ρ_{k}^{3/2}}{4πμ_{k}^{3/2}} δ_{pq} + \frac{(κ_{0} − κ)ρ_{k}^{3/2}}{12π} [\frac{1}{(λ_{k} + 2μ_{k})^{3/2}} − \frac{1}{μ_{k}^{3/2}} ]δ_{pq} =

= \frac{(κ_{0} − κ)ρ_{k}^{3/2}}{12π} [\frac{1}{(λ_{k} + 2μ_{k})^{3/2}} + \frac{2}{μ_{k}^{3/2}} ]δ_{pq}, p, q = \overline{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 problem of the operator A^{k}(∂x) − κ_{0}^{2}I ” to the 3 × 3 matrix G(x, y, −κ_{0}^{2}) =

G^{k}(x, y, −κ_{0}^{2}), x ∈ D_{k}, y ∈ D (D = ∪_{k=0}^{m_{0}}D_{k}), x ≠ y, k ∈ \overline{0, m_{0}} which satisfies

the following conditions:

(1) ∀x∈D_{k}, ∀y ∈D, x ≠ y :

A^{k}(∂x)G^{k}(x, y, −κ_{0}^{2})−κ_{0}^{2}G^{k}(x, y, −κ_{0}^{2})=0, k =\overline{0, m_{0}},

(2) ∀z ∈ S_{k}, ∀y ∈ D : G^{∘}^{+}(x, y, −κ_{0}^{2}) = G^{k} − (x, y, −κ_{0}^{2}),

(T^{∘} (∂_{z}, n(z))G^{∘}(z, y, −κ_{0}^{2}))^{+} = (T^{k} (∂_{z}, n(z))G^{k}(z, y, −κ_{0}^{2}))^{−}, k = \overline{1, m_{0}},

(3) ∀z ∈ S_{k}, ∀y ∈ D : G^{∘}^{+}(z, y, −κ_{0}^{2}) = 0, k = 0, m_{0} + 1, …, m;

(4) G^{k}(x, y, −κ_{0}^{2})=Γ^{k}(x − y, −κ_{0}^{2})−g^{k}(x, y, −κ_{0}^{2}), x∈D_{k}, y ∈D, k =\overline{0, m_{0}},

where g^{k}(x, y, −κ_{0}^{2}) is a regular in D_{k} solution of the following problem:

(1) ∀x ∈ D_{k}, ∀y ∈ D : A^{k}(∂x)g^{k}(x, y, −κ_{0}^{2})−κ_{0}^{2}g^{k}(x, y, −κ_{0}^{2}) = 0, k = \overline{0, m_{0}};

(2) ∀z ∈ S_{k}, ∀y ∈ D : g^{∘}^{+}(z, y, −κ_{0}^{2}) − g^{k}^{−}(z, y, −κ_{0}^{2}) =

= Γ^{∘}(z − y, −κ_{0}^{2}) − Γ^{k}(z − y, −κ_{0}^{2}),

(T^{∘} (∂_{z}, n(z))g^{∘}(z, y, −κ_{0}^{2}))^{+} − (T^{k} (∂_{z}, n(z))g^{k}(z, y, −κ_{0}^{2}))^{−} =

= T^{∘} (∂_{z}, n(z))Γ^{∘}(z − y, −κ_{0}^{2}) − T^{k} (∂_{z}, n(z))Γ^{k}(z − y, −κ_{0}^{2}), k = \overline{1, m_{0}};

(3) ∀z ∈ S_{k}, ∀y ∈ D : g^{∘}^{+}(z, y, −κ_{0}^{2}) = Γ^{∘}(z − y, −κ_{0}^{2}),

k = 0, m_{0} + 1, …, m; The solvability of this problem is shown in [8] and thereby the existence of

G(x, y, −κ_{0}^{2}) is proved. As is known [8], G(x, y, −κ_{0}^{2}) possesses a symmetry property of the form

G(x, y, −κ_{0}^{2}) = G^{T} (y, x, −κ_{0}^{2}).  (11)

Moreover, we have the estimates [9]

\begin{matrix}∀(x, y) ∈ D_{k} × D_{k} : G_{pq}(x, y, −κ_{0}^{2}) = O(|x − y|^{−1}), \\ \frac{∂}{∂x_{j}} G_{pq}(x, y, −κ_{0}^{2}) = O(|x − y|^{−2}), m, n, j = \overline{1, 3}; k = \overline{0, m_{0}}. \end{matrix}\} (12)

7. Let u(x) = u^{k}(x) and v(x) = v^{k}(x), x ∈ D_{k}, be arbitrary (regular) vectors of the class C^{1}(\overline{D}_{k}) ∩ C^{2}(D_{k}), k = \overline{0, m_{0}}. Then the following Green formula is valid [8]:

∑_{k=0}^{m_{0}} ∫_{D_{k}} (v^{k}A^{k}u^{k} + E^{k}(v^{k}, u^{k}))dx = ∫_{S} v^{∘}^{+}(T^{∘} u^{∘})^{+}ds +

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} (v^{∘}^{+}(T^{∘} u^{∘})^{+} − v^{k}^{−}(T^{k} u^{k})^{−})ds,  (13)

where S = S_{0} ∪_{k=m_{0}+1}^{m} S_{k},

E^{k}(v^{k}, u^{k}) = \frac{ρ_{k}^{−1}(3λ_{k} + 2μ_{k})}{3} div v^{k} div u^{k} + \frac{ρ_{k}^{−1}μ_{k}}{2} ∑_{p≠q} (\frac{∂v^{k}_{p}}{∂x_{q}} +

+\frac{∂v^{k}_{q}}{∂x_{p}} )(\frac{∂u^{k}_{p}}{∂x_{q}} + \frac{∂u^{k}_{q}}{∂x_{p}} ) + \frac{ρ_{k}^{−1}μ_{k}}{3} ∑_{p,q} (\frac{∂v^{k}_{p}}{∂x_{p}} − \frac{∂v^{k}_{q}}{∂x_{q}} )(\frac{∂u^{k}_{p}}{∂x_{p}} − \frac{∂u^{k}_{q}}{∂x_{q}} ). (14)

It follows from (14) that E^{k}(v^{k}, u^{k}) = E^{k}(u^{k}, v^{k}) and E^{k}(v^{k}, v^{k}) ≥ 0.

For the regular in D_{k}, k = \overline{0, m_{0}} vector u(x) the following general integral representation is valid [8]:

∀y ∈ D_{k} : u_{j}(y) = − ∑_{k=0}^{m_{0}} ∫_{D_{k}} Γ^{k}_{j}(x − y, −κ^{2})(A^{k}(∂x)u^{k}(x) − κ^{2}u^{k}(x))dx +

+ ∫_{S} [Γ^{k}_{j}(z − y, −κ^{2})(T^{∘} (∂_{z}, n(z))u^{∘}(z))^{+} −

−u^{∘}^{+}(z)T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2})]d_{z}S +

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} [Γ^{∘}_{j}(z − y, −κ^{2})(T^{∘} (∂_{z}, n(z))u^{∘}(z))^{+} −

−Γ^{k}_{j}(z − y, −κ^{2})(T^{k} (∂_{z}, n(z))u^{k}(z))^{−}]d_{z}S −

− ∑_{k=1}^{m_{0}} ∫_{S_{k}} [u^{∘}^{+}(z)(T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2}) −

−u^{k}^{−}(z)(T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S, j = 1, 2, 3.  (15)

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

L[u] = ∑_{k=0}^{m_{0}} ∫_{D_{k}} (E^{k}(u^{k}, u^{k}) + κ^{2}u^{k}^{2})dx − 2 ∑_{k=1}^{m_{0}} ∫_{S_{k}} [u^{∘}^{+}(z)T^{∘} (∂_{z}, n(z)) ×

×Γ^{∘}_{j}(z − y, −κ^{2}) − u^{k}^{−}(z)(T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S,  (16)

where j = 1, 2, 3 is the fixed number and y is an arbitrary fixed point in D_{k}, k = \overline{0, m_{0}}, which is defined in the class of regular in D_{k}, k = \overline{0, m_{0}}, vector functions satisfying the conditions:

(1) ∀z ∈ S_{k} : u^{∘}^{+}(z) − u^{k}^{−}(z) = Γ^{∘}_{j}(z − y, −κ^{2}) − Γ^{k}_{j}(z − y, −κ^{2}),

(T^{∘} u^{∘}(z))^{+} − (T^{k} u^{k}(z))^{−} = T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2})−

T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2}), k = \overline{1, m_{0}},

(2) ∀z ∈ S_{k} : u^{∘}^{+}(z) = Γ^{∘}_{j}(z − y, −κ^{2}), k = 0, m_{0} + 1, …, m.

Theorem 1. The functional L takes a minimal value for u=g_{j}(x, y,−κ^{2}).

Proof. Let u be an arbitrary vector from the domain of definition of the functional L, and let v = u − g_{j}. Then, with (14) taken into account, (16) implies

L[u] = L[v + g_{j}] = ∑_{k=0}^{m_{0}} ∫_{D_{k}} (E^{k}(v^{k} + g^{k}_{j}, v^{k} + g^{k}_{j}) + κ^{2}(v^{k} + g^{k}_{j})^{2}]dx −

−2 ∑_{k=1}^{m_{0}} ∫_{S_{k}} [(v^{∘} + g^{∘}_{j})T^{∘} Γ^{∘}_{j} − (v^{k}^{−} + g^{k}_{j}^{−})T^{k} Γ^{k}_{j}]ds =

= ∑_{k=0}^{m_{0}} ∫_{D_{k}} [E^{k}(v^{k}, v^{k}) + 2E^{k}(v^{k}, g^{k}_{j}) + E^{k}(g^{k}_{j}, g^{k}_{j}) + κ^{2}(v^{k}^{2} + 2v^{k}g^{k}_{j} + g^{k}_{j}^{2})]dx −

− ∑_{k=1}^{m_{0}} ∫_{S_{k}} [v^{∘}^{+}T^{∘} Γ^{∘}_{j} − v^{k}^{−}T^{k} Γ^{k}_{j}]ds − 2 ∑_{k=1}^{m_{0}} ∫_{S_{k}} (g^{∘}_{j}^{+}T^{∘} Γ^{∘}_{j} − g^{k}_{j}^{−}T^{k} Γ^{k}_{j}]ds =

= L[g_{j}] + ∑_{k=0}^{m_{0}} ∫_{D_{k}} [E^{k}(v^{k}, v^{k}) + κ^{2}v^{k}^{2}]dx + 2 ∑_{k=0}^{m_{0}} ∫_{D_{k}} [E^{k}(v^{k}, g^{k}_{j}) +

+κ^{2}v^{k}g^{k}_{j}]dx − 2 ∑_{k=1}^{m_{0}} ∫_{S_{k}} [v^{∘}^{+}T^{∘} Γ^{∘}_{j} − v^{k}^{−}T^{k} Γ^{k}_{j}]ds.  (17)

Using the Green formula (13) for v = u − g_{j}, u = g_{j} and taking into

account that A^{k}g^{k}_{j} = κ^{2}g^{k}_{j}, v^{∘}^{+}(z) = 0 for z ∈ S_{k}, k = 0, m_{0} + 1, …, m, we

obtain

∑_{k=0}^{m_{0}} ∫_{D_{k}} [E^{k}(v^{k}, g^{k}_{j}) + κ^{2}v^{k}g^{k}_{j}]dx = ∑_{k=1}^{m_{0}} ∫_{S_{k}} [v^{∘}^{+}(T^{∘} g^{∘}_{j})^{+} − v^{k}^{−}(T^{k} g^{k}_{j})^{−}]ds. (18)

Now, since v^{∘}^{+}(z) = v^{k}^{−}(z), (T^{∘} g^{∘}_{j})^{+} − (T^{k} g^{k}_{j})^{−} = T^{∘} Γ^{∘}_{j} − T^{k} Γ^{k}_{j} for z ∈ S_{k},

k = \overline{1, m_{0}} (17) by virtue of (18) takes the form

L[u] = L[g_{j}] + ∑_{k=0}^{m_{0}} ∫_{D_{k}} [E^{k}(v^{k}, v^{k}) + κ^{2}v^{k}^{2}]dx ≥ L[g_{j}].

Theorem 2. The estimate

|g_{jj}(y, y, −κ^{2}) − g_{jj}(y, y, −κ_{0}^{2})| ≤ \frac{const}{l_{y}^{1+δ}}, y ∈ D, δ > 0, (19)

holds for the function g_{jj}(y, y, −κ^{2}).

Proof. We write formula (15) for u_{j}(x)=g_{jj}(x, y, −κ^{2}) and Γ_{j}(x−y, −κ^{2})= G_{j}(x − y, −κ^{2}). Then taking into account the boundary and contact conditions for g and G, we get

∀(x, y) ∈ D_{k} : g_{jj}(x, y, −κ^{2}) =

= − ∫_{S} Γ^{∘}_{j}(z − x, −κ^{2})(T^{∘} (∂_{z}, n(z))G^{∘}_{j}(z, y, −κ^{2}))^{+}d_{z}S −

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} [G^{∘}_{j}^{+}(z, x, −κ^{2})T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2}) −

−G^{k}_{j}^{−}(z, x, −κ^{2})T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S −

− ∑_{k=1}^{m_{0}} ∫_{S_{k}} [Γ^{∘}_{j}(z − x, −κ^{2})(T^{∘} (∂_{z}, n(z))G^{∘}_{j}(z, y, −κ^{2}))^{+} −

−Γ^{k}_{j}(z − x, −κ^{2})(T^{k} (∂_{z}, n(z))G^{k}_{j}(z, y, −κ^{2}))^{−}]d_{z}S.  (20)

Using formula (13) for u = v, (16) can be rewritten as

L[u] = − ∑_{k=0}^{m_{0}} ∫_{D_{k}} u^{k}[A^{k}(∂_{x})u^{k}(x) − κ^{2}u^{k}(x)]dx + ∫_{S} u^{∘}^{+}(z)(T^{∘} u^{∘}(z))^{+}d_{z}S +

116 T. BURCHULADZE AND R. RUKHADZE

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} [u^{∘}^{+}(z)(T^{∘} u^{∘}(z))^{+} − u^{k}^{−}(z)(T^{k} u^{k}(z))^{−}]d_{z}S −

−2 ∑_{k=1}^{m_{0}} ∫_{S_{k}} [u^{∘}^{+}(z)T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2}) −

−u^{k}^{−}(z)T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S.  (21)

which for u(x) = g_{j}(x, y, −κ^{2}) = Γ_{j}(x − y, −κ^{2}) − G_{j}(x, y, −κ^{2}) implies

L[g_{j}] = ∫_{S} Γ^{∘}_{j}(z − y, −κ^{2})T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2})d_{z}S −

− ∫_{S} Γ^{∘}_{j}(z − y, −κ^{2})(T^{∘} (∂_{z}, n(z))G^{∘}_{j}(z, y, −κ^{2}))^{+}d_{z}S +

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} [Γ^{∘}_{j}(z − y, −κ^{2})T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2}) −

−Γ^{k}_{j}(z − y, −κ^{2})T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S +

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} [G^{∘}_{j}^{+}(z, y, −κ^{2})T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2}) −

−G^{k}_{j}^{−}(z, y, −κ^{2})T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S −

− ∑_{k=1}^{m_{0}} ∫_{S_{k}} [Γ^{∘}_{j}(z − y, −κ^{2})(T^{∘} (∂_{z}, n(z))G^{∘}_{j}(z, y, −κ^{2}))^{+} −

−Γ^{k}_{j}(z − y, −κ^{2})(T^{k} (∂_{z}, n(z))G^{k}_{j}(z, y, −κ^{2}))^{−}]d_{z}S.  (22)

On the basis of (22), from (20) we obtain

g_{jj}(y, y, −κ^{2}) = L[g_{j}] − ∫_{S} Γ^{∘}_{j}(z − y, −κ^{2})T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2})d_{z}S +

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} [Γ^{∘}_{j}(z − y, −κ^{2})T^{∘} (∂_{z}, n(z))Γ^{∘}_{j}(z − y, −κ^{2}) −

−Γ^{k}_{j}(z − y, −κ^{2})T^{k} (∂_{z}, n(z))Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S.  (23)

The vector _{k} ; Γ_{j}(x − y, −κ^{2}) defined by (8) belongs to the domain of definition of the functional L and, since g_{j}(x, y, −κ^{2}) imparts a minimal value to the functional L, it is obvious that

L[g_{j}] ≤ L[Γ̂_{j}].

Now (23) implies

g_{jj}(y, y, −κ^{2}) ≤ L[Γ̂_{j}] − ∫_{S} Γ^{∘}_{j}T^{∘} Γ^{∘}_{j}ds +

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} (Γ^{∘}_{j}T^{∘} Γ^{∘}_{j} − Γ^{k}_{j}T^{k} Γ^{k}_{j})ds, y ∈ D_{k}.  (24)

By virtue of the properties of Γ̂, from (21) we obtain

L[Γ̂_{j}] = − ∫_{B(y,l_{y})} Γ̂_{j}(AΓ̂_{j} − κ^{2}Γ̂_{j})dx + ∫_{S} Γ^{∘}_{j}T^{∘} Γ^{∘}_{j}ds −

− ∑_{k=1}^{m_{0}} ∫_{S_{k}} (Γ^{∘}_{j}T^{∘} Γ^{∘}_{j} − Γ^{k}_{j}T^{k} Γ^{k}_{j})ds.  (25)

By (25) and (24) we have

g_{jj}(y, y, −κ^{2}) ≤ − ∫_{B(y,l_{y})}Γ̂_{j}(AΓ̂_{j} − κ^{2}Γ̂_{j})dx, y ∈ D_{k}, k = \overline{0, m_{0}}. (26)

Taking into account estimates (9), for m = 5 we obtain

|Γ̂_{mj}(x, y, −κ^{2})| ≤ \frac{const}{l_{y}} for s = 0,

|κ^{2}Γ̂_{ij}(x, y, −κ^{2})| ≤ κ^{2} \frac{const e^{−ακr}}{l_{y}^{5}} r^{4} =

= \frac{const}{l_{y}^{5}}r^{2}(κ^{2}r^{2})e^{−ακr} ≤ \frac{const}{l_{y}^{3}} for s = 0,

|AΓ̂_{j}(x, y, −κ^{2})| ≤ \frac{const}{l_{y}^{3}} for s = 2.

Hence (26) implies

g_{jj}(y, y, −κ^{2}) ≤ \frac{const}{l_{y}^{4}} · \frac{4}{3}πl_{y}^{3} ≤ \frac{const}{l_{y}} ≤ \frac{const}{l_{y}^{1+δ}},  (27)

where δ > 0 is an arbitrary integer.

Let us estimate g_{jj}(y, y, −κ^{2}) from below. For this we introduce the notation:

M[u] = ∑_{k=0}^{m_{0}} ∫_{D_{k}} [E^{k}(u^{k}, u^{k}) + κ^{2}u^{k}^{2}]dx, M_{0}[u] = ∑_{k=0}^{m_{0}} ∫_{D_{k}} [E^{k}(u^{k}, u^{k}) + κ_{0}^{2}u^{k}^{2}]dx,

N[u] = ∑_{k=1}^{m_{0}} ∫_{S_{k}} [u^{∘}^{+}(z)T^{∘} Γ^{∘}_{j}(z − y, −κ^{2}) − u^{k}^{−}(z)T^{k} Γ^{k}_{j}(z − y, −κ^{2})]d_{z}S.

Since κ_{0}^{2} ≤ κ^{2}, we have

L[g_{j}(x, y, −κ^{2})] = min L[u] = min(M[u] − 2N[u]) ≥ min(M_{o}[u] − 2N[u]).

Let the vector function φ(x, y) impart a minimal value to the functional M_{0}[u] − 2N[u]. Then φ(x, y) is a regular in D_{k} (k = \overline{0, m_{0}}) solution of the following problem:

(1) ∀x ∈ D_{k}, ∀y ∈ D : A^{k}(∂x)φ^{k}(x, y) − κ_{0}^{2}φ^{k}(x, y) = 0, k = \overline{0, m_{0}};

(2) ∀z ∈ S_{k}, ∀y ∈ D : φ^{∘}^{+}(z, y)−φ^{k}^{−}(z, y) = Γ^{∘}(z−y, −κ^{2})−Γ^{k}(z−y, −κ^{2}),

(T^{∘} φ^{∘}(z, y))^{+} − (T^{k} φ^{k}(z, y))^{−} = T^{∘} Γ^{∘}(z − y, −κ^{2}) − T^{k} Γ^{k}(z − y, −κ^{2});

(3) ∀z ∈ S_{k}, ∀y ∈ D : φ^{∘}^{+}(z, y) = Γ^{∘}(z − y, −κ^{2}), k = 0, m_{0} + 1, …, m.

Rewriting formula (15) for φ(x, y) with Γ = G, we obtain

∀(x, y) ∈ D_{k} : φ(x, y) = − ∫_{S} Γ^{∘}_{j}(z − x, −κ^{2})T^{∘} G^{∘}_{j}(z, y, −κ_{0}^{2})d_{z}s +

+ ∑_{k=1}^{m_{0}} ∫_{S_{k}} [G^{∘}_{j}(z, x, −κ_{0}^{2})T^{∘} Γ^{∘}_{j}(z, y, −κ^{2}) − G^{k}_{j}^{−}(z, x, −κ_{0}^{2}) ×

×T^{k} Γ^{k}_{j}(z − y, −κ^{2})]d_{z}s − ∑_{k=1}^{m_{0}} ∫_{S_{k}} [Γ^{∘}_{j}(z − x, −κ^{2})(T^{∘} G^{∘}_{j}(z, y, −κ_{0}^{2}))^{+} −

−Γ^{k}_{j}(z − x, −κ^{2})(T^{k} G^{k}_{j}(z, y, −κ_{0}^{2}))]d_{z}s.  (28)

By (7) and (12) and the theorem on kernel composition [10] it follows from (28) that

∀(x, y) ∈ \overline{D}_{k} × D_{k} : |φ(x, y)| ≤ \frac{const}{r_{xy}}, k = \overline{0, m_{0}}, r_{xy} = |x − y|. (29)

In that case

L[g_{j}(x, y, −κ^{2}) ≥ M_{0}[φ] − 2N[φ] ≥ −2N[φ] =

=−2 ∑_{k=1}^{m_{0}} ∫_{S_{k}} [φ^{∘}^{+}(z, y)T^{∘} Γ^{∘}_{j}(z−y,−κ^{2}) − φ^{k}^{−}(z, y)T^{k} Γ^{k}_{j}(z−y,−κ^{2})]d_{z}s. (30)

Now taking into account that

∀(z, y) ∈ S_{k} × D_{k} : |φ(z, y)| ≤ \frac{const}{r_{zy}} ≤ \frac{1}{l_{y}} ,

∀(z, y) ∈ S_{k} × D_{k} : |T Γ_{j}(z − y, −κ^{2})| ≤ \frac{const}{r_{zy}^{2}} = \frac{const}{r_{zy}^{δ}r_{zy}^{2−δ}} ≤

≤ \frac{1}{l_{y}^{δ}} · \frac{1}{r_{zy}^{2−δ}} , δ > 0,

from (30) we obtain

L[g_{j}] ≥ −\frac{const}{l_{y}^{1+δ}}, δ > 0.

By virtue of representation (23) we can easily conclude that the estimate

∀y ∈ D : g_{jj}(y, y, −κ^{2}) ≥ −\frac{const}{l_{y}^{1+δ}}  (31)

holds.

(27) and (31) imply (19).

9. Consider the first boundary-contact problem on eigenvalues: In the domain D_{k} (k = \overline{0, m_{0}}), find a regular vector w(x) = w^{k}(x), x ∈ D_{k}, which is a nontrivial solution of the equations

∀x ∈ D_{k} : A^{k}(∂x)w^{k}(x) + γw^{k}(x) = 0, k = \overline{0, m_{0}},

satisfying the contact conditions

∀z ∈ S_{k} : w^{∘}^{+}(z) = w^{k}^{−}(z), (T^{∘} w^{∘}(z))^{+} = (T^{k} w^{k}(z))^{−}, k = \overline{1, m_{0}},

and the boundary condition

∀z ∈ S_{k} : w^{∘}^{+}(z) = 0, k = 0, m_{0} + 1, …, m.

We denote this problem by I_{γ}. In the same manner as in [8] we can show that problem I_{γ} is equivalent to a system of integral equations

w(x) = (γ + κ_{0}^{2}) ∫_{D} G(x, y, −κ_{0}^{2})w(y)dy.  (32)

By virtue of (11) and (12) equation (32) is an integral equation with a symmetric kernel of the class L_{2}(D). Hence it follows that there exists a countable system of eigenvalues (γ_{n} + κ_{0}^{2})_{n=1}^{∞} and the corresponding

orthonormal in D system of eigenvectors {w^{(n)}(x)}_{n=1}^{∞} = {w^{k}^{(n)}(x)}_{n=1}^{∞}, x ∈ D_{k}, k = \overline{0, m_{0}}, of equation (32). Therefore (γ_{n})_{n=1}^{∞} and \{w^{(n)}(x)\}_{n=1}^{∞} are respectively the eigenvalues and eigenvectors of problem I_{γ}. As established in [8], all γ_{n} > 0. Moreover, it is proved in [11] that the system {w^{(n)}(x)}_{n=1}^{∞} is complete in L_{2}(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

∫_{0}^{∞} \frac{dΦ(t)}{(x + t)^{l}} ∼ \frac{p}{x^{m}} ,

where l, m, p satisfy the conditions 0 < m < l, p ≠ 0, is fulfilled, then

Φ(t) ∼ \frac{pΓ(l)}{Γ(m)Γ(l − m + 1)}t^{l−m}.

Here Γ is the Euler function.

Write the kernel expansion in terms of the eigenfunctions

G(x, y, −κ^{2}) = ∑_{n=1}^{∞} \frac{w^{(n)}(x) × w^{(n)}(y)}{γ_{n} + κ^{2}},  (33)

G(x, y, −κ_{0}^{2}) = ∑_{n=1}^{∞} \frac{w^{(n)}(x) × w^{(n)}(y)}{γ_{n} + κ_{0}^{2}},  (34)

where x, y ∈ D_{k}, k = \overline{0, m_{0}}, and the symbol × denotes the matrix product of a column vector by a row vector (dyadic product) w^{(n)}(x) × w^{(n)}(y) = ∥w_{i}^{(n)}(x) · w_{j}^{(n)}(y)∥ i, j = 1, 2, 3. From (33) and (34) we obtain

G(x, y, −κ^{2}) − G(x, y, −κ_{0}^{2}) = (κ_{0}^{2} − κ^{2}) ∑_{n=1}^{∞} \frac{w^{(n)}(x) × w^{(n)}(y)}{(γ_{n} + κ^{2})(γ_{n} + κ_{0}^{2})}. (35)

Passage in equality (35) to the limit as x → y results in

(κ_{0}^{2} − κ^{2}) ∑_{n=1}^{∞} \frac{[w_{j}^{(n)}(y)]^{2}}{(γ_{n} + κ^{2})(γ_{n} + κ_{0}^{2})} = lim_{x→y} [Γ_{jj}(x − y, −κ^{2}) −

−Γ_{jj}(x − y, −κ_{0}^{2})] − lim_{x→y} [g_{jj}(x, y, −κ^{2}) − g_{jj}(x, y, −κ_{0}^{2})], (36)

ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS 121

x, y ∈ D_{k}, k = \overline{0, m_{0}}, j = \overline{1, 3}.

Taking into account (10) and (19), we find from (36) that

∑_{n=1}^{∞} \frac{[w_{j}^{(n)}(y)]^{2}}{(γ_{n} + κ^{2})(γ_{n} + κ_{0}^{2})} ∼ \frac{ρ_{k}^{3/2}}{12π(κ + κ_{0})}[\frac{1}{(λ_{k} + 2μ_{k})^{3/2}} + \frac{2}{μ_{k}^{3/2}} ], (37)
y ∈ D_{k}, k = \overline{0, m_{0}}, j = \overline{1, 3}.

Consider the function

Φ_{j}(t) = ∑_{γ_{n}≤t} \frac{[w_{j}^{(n)}(y)]^{2}}{γ_{n} + κ_{0}^{2}} y ∈ D_{k}, k = \overline{0, m_{0}}, j = \overline{1, 3}.

It is easy to observe that

∫_{0}^{∞} \frac{dΦ_{j}(t)}{t + κ^{2}} = ∑_{n=1}^{∞} \frac{[w_{j}^{(n)}(y)]^{2}}{(γ_{n} + κ^{2})(γ_{n} + κ_{0}^{2})}.

By (37) we have

∫_{0}^{∞} \frac{dΦ_{j}(t)}{t + κ^{2}} ∼ \frac{A_{k}}{κ} ,  (38)

where

A_{k} = \frac{ρ_{k}^{3/2}}{12π} [\frac{1}{(λ_{k} + 2μ_{k})^{3/2}} + \frac{2}{μ_{k}^{3/2}} ].

By Theorem 3 with regard for (38) we obtain

Φ_{j}(t) = ∑_{γ_{n}≤t} \frac{[w_{j}^{(n)}(y)]^{2}}{γ_{n} + κ_{0}^{2}} ∼ \frac{A_{k}Γ(1)}{Γ(\frac{1}{2})Γ(\frac{3}{3})}t^{1/2} = \frac{2}{π} A_{k}t^{1/2}.  (39)

Obviously,

∑_{γ_{n}≤t}[w_{j}^{(n)}(y)]^{2} = ∫_{0}^{t}(ξ + κ_{0}^{2})dΦ_{j}(ξ) = (ξ + κ_{0}^{2})Φ_{j}(ξ)|_{0}^{t} − ∫_{0}^{t} Φ_{j}(ξ)dξ.

Taking into account (39), we obtain

∑_{γ_{n}≤t}[w_{j}^{(n)}(y)]^{2} ∼ \frac{2}{3π} A_{k}t^{3/2}, y ∈ D_{k}, k = \overline{0, m_{0}}, j = \overline{1, 3}.

If we sum these relations with respect to j = 1, 2, 3, then we shall have

∑_{γ_{n}≤t}[w_{j}^{(n)}(y)]^{2} ∼ \frac{2}{π} A_{k}t^{3/2}, y ∈ D,

or

∑_{γ_{n}≤t}[w^{k}_{j}^{(n)}(y)]^{2} ∼ \frac{ρ_{k}^{3/2}}{6π^{2}} [\frac{1}{(λ_{k}+2μ_{k})^{3/2}} + \frac{2}{μ_{k}^{3/2}} ]t^{3/2}, y ∈ D_{k}, k = \overline{0, m_{0}}. (40)

The above expression provides the asymptotic distribution of eigenfunctions. 11. Taking into account (10), from (36) we obtain

(κ^{2} − κ_{0}^{2}) ∑_{n=1}^{∞} \frac{[w^{(n)}(y)]^{2}}{(γ_{n} + κ_{0}^{2})(γ_{n} + κ^{2})} = 3(κ − κ_{0})A_{k} +

+ ∑_{j=1}^{3} [g_{jj}(y, y, −κ_{0}^{2}) − g_{jj}(y, y, −κ^{2})].  (41)

Denote

φ(y, κ) = 3(κ − κ_{0})A_{k} + ∑_{j=1}^{3} [g_{jj}(y, y, −κ_{0}^{2}) − g_{jj}(y, y, −κ^{2})].

Now (41) yields

\frac{φ(y, κ)}{κ^{2} − κ_{0}^{2}} = ∑_{n=1}^{∞} \frac{[w^{(n)}(y)]^{2}}{(γ_{n} + κ_{0}^{2})(γ_{n} + κ^{2})} ≤ ∑_{n=1}^{∞} \frac{[w^{(n)}(y)]^{2}}{(γ_{n} + κ_{0}^{2})^{2}} .  (42)

By virtue of the Bessel inequality we have

∑_{n=1}^{∞} \frac{[w^{(n)}(x)]^{2}}{(γ_{n} + κ_{0}^{2})^{2}} ≤ ∫_{D} |G(x, y, −κ_{0}^{2})|^{2}dy, x ∈ D_{k}, k = \overline{0, m_{0}}. (43)

By estimate (8) it follows from (42) that the sum of the series

∑_{n=1}^{∞} \frac{[w^{(n)}(x)]^{2}}{(γ_{n} + κ_{0}^{2})^{2}}

exists and is uniformly bounded in \overline{D}_{k}. This and (42) imply

∀y ∈ D_{k} : |φ(y, κ)| ≤ const(κ^{2} − κ_{0}^{2}), k = \overline{0, m_{0}}.  (44)

Integrating equality (42) in D and using the fact that the vectors [w^{(n)}(x)]_{n=1}^{∞} are orthonormal in D, we obtain

∫_{D} φ(y, κ)dy = (κ^{2} − κ_{0}^{2}) ∑_{n=1}^{∞} \frac{1}{(γ_{n} + κ_{0}^{2})(γ_{n} + κ^{2})},  (45)

∫_{D} φ(y, κ)dy = ∫_{D} 3(κ − κ_{0})A_{k}dy + ∫_{D} ∑_{j=1}^{3} [g_{jj}(y, y, −κ_{0}^{2}) −

ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS 123

−g_{jj}(y, y, −κ^{2})]dy = 3(κ − κ_{0}) ∑_{k=0}^{m_{0}} A_{k} mes D_{k} +

+ ∑_{j=1}^{3} ∫_{D} [g_{jj}(y, y, −κ_{0}^{2}) − g_{jj}(y, y, −κ^{2})]dy.  (46)

By (46), from (45) we have

∑_{n=1}^{∞} \frac{1}{(γ_{n} + κ_{0}^{2})(γ_{n} + κ^{2})} − \frac{3}{κ + κ_{0}} ∑_{k=0}^{m_{0}} A_{k} mes D_{k} =

= \frac{1}{κ^{2} − κ_{0}^{2}} ∑_{j=1}^{3} ∫_{D} [g_{jj}(y, y, −κ_{0}^{2}) − g_{jj}(y, y, −κ^{2})]dy.  (47)

Denote by D_{kη} that part of D_{k}, k = \overline{0, m_{0}}, whose points lie from the boundary D_{k} at a distance less than η. D_{η} = ∪_{k=0}^{m_{0}}D_{kη}. Then

∫_{D} [g_{jj}(y, y, −κ_{0}^{2}) − g_{jj}(y, y, −κ^{2})]dy = ∫_{D\D_{η}} [g_{jj}(y, y, −κ_{0}^{2}) −

−g_{jj}(y, y, −κ^{2})]dy + ∫_{D_{η}} φ(y, κ)dy − ∫_{D_{η}} 3(κ − κ_{0})A_{k}dy.

This and (47) imply

| ∑_{n=1}^{∞} \frac{1}{(γ_{n} + κ_{0}^{2})(γ_{n} + κ^{2})} − \frac{3}{κ + κ_{0}} ∑_{k=0}^{m_{0}} A_{k} mes D_{k}| ≤

≤ \frac{1}{κ^{2} − κ_{0}^{2}} | ∫_{D_{η}} φ(y, κ)dy| + \frac{1}{κ^{2} − κ_{0}^{2}} ∑_{j=1}^{3} | ∫_{D\D_{η}} [g_{jj}(y, y, −κ_{0}^{2}) −

−g_{jj}(y, y, −κ)]dy| + \frac{3}{κ + κ_{0}} | ∫_{D_{m}} A_{k}dy|.  (48)

The following estimates are valid:

\frac{3}{κ + κ_{0}} | ∫_{D_{η}} A_{k}dy| = \frac{3}{κ + κ_{0}} | ∑_{k=0}^{m_{0}} ∫_{D_{kη}} A_{k}dy| ≤ const \frac{3}{κ + κ_{0}} η, (49)
\frac{1}{κ^{2} − κ_{0}^{2}} | ∫_{D_{η}} φ(y, κ)dy| ≤ \frac{const}{κ^{2} − κ_{0}^{2}} (κ^{2} − κ_{0}^{2})η = const η,  (50)

\frac{1}{κ^{2}−κ_{0}^{2}} ∑_{j=1}^{3} | ∫_{D\D_{η}} [g_{jj}(y, y,−κ_{0}^{2})−g_{jj}(y, y,−κ)]dy|≤ \frac{const}{κ^{2}−κ_{0}^{2}} \frac{1}{η^{δ}} . (51)

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

Φ(t) = ∑_{γ_{n}≤t} \frac{1}{γ_{n} + κ_{0}^{2}} .

It is easy to verify that

∫_{0}^{∞} \frac{dΦ(t)}{t + κ^{2}} = ∑_{n=1}^{∞} \frac{1}{(γ_{n} + κ_{0}^{2})(γ_{n} + κ^{2})}.

By of (49), (50) and (51) we find from (48) that

∫_{0}^{∞} \frac{dΦ(t)}{t + κ^{2}} ∼ \frac{3 ∑_{k=0}^{m_{0}} A_{k} mes D_{k}}{κ}

for η = \frac{1}{κ^{2}−κ_{0}^{2}} . According to Theorem 3 we have

Φ(t) ∼ \frac{6 ∑_{k=0}^{m_{0}} A_{k} mes D_{k}}{π} t^{1/2}.  (52)

Denoting

N(t) = ∑_{γ_{n}≤t} 1,

for a number of eigenvalues not higher than t, we get

N(t) = ∫_{0}^{t}(ξ + κ_{0}^{2})dΦ(ξ) = (ξ + κ_{0}^{2})Φ(ξ)|_{0}^{t} − ∫_{0}^{t} Φ(ξ)dξ.

Hence with regard for (52) we obtain

N(t) ∼ \frac{2}{π} ∑_{k=0}^{m_{0}} A_{k} mes D_{k} · t^{3/2},

or, finally,

N(t) ∼ \frac{1}{6π^{2}} ∑_{k=0}^{m_{0}} [\frac{1}{(λ_{k} + 2μ_{k})^{3/2}} + \frac{2}{μ_{k}^{3/2}} ]ρ_{k}^{3/2} mes D_{k} · t^{3/2}. (53)

Thus the results of this paper can be formulated as

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 S_{k} (k = 0, m_{0} + 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 S_{k} (k = 0, m_{0} + 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 λ_{k} = λ, μ_{k} = μ, ρ_{k} = ρ (k = \overline{0, m_{0}}) i.e., if a homogeneous elastic medium D_{0} ∪ ( ∪_{k=1}^{m_{0}}D_{k}) ∪ ( ∪_{k=1}^{m_{0}}S_{k}) is considered, then, as can readily

be verified, [w^{(n)}(x)]_{n=1}^{∞} and (γ_{n})_{n=1}^{∞} will be the eigenvector-functions and 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

N(t) ∼ \frac{1}{4π} ∑_{k=0}^{m_{0}} [\frac{1}{λ_{k} + 2μ_{k}} + \frac{1}{μ_{k}} ]ρ_{k} mes D_{k} · t,

where mes D_{k} is a space of the domain D_{k}.

References

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3. A. Plejel, Proprietes asymptotiques des fonctions et vareurs propres de certains problems de vibrations. Arkiv für Math., Astr. Och Fysik. 27 A(1940), 1–100.

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(Received 15.05.1997)

Authors’ addresses:

Tengiz Burchuladze Roland Rukhadze A. Razmadze Mathematical Institute Georgian Technical University Georgian Academy of Sciences Department of Mathematics (3) 1, M. Aleksidze St., Tbilisi 380093 77, M. Kostava St., Tbilisi 380075 Georgia Georgia