(1)ASYMPTOTIC DISTRIBUTION OF EIGENFUNCTIONS AND EIGENVALUES OF THE BASIC BOUNDARY-CONTACT OSCILLATION PROBLEMS OF THE CLASSICAL THEORY OF ELASTICITY T

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 con- sidered for a three-dimensional piecewise-homogeneous isotropic elas- tic medium bounded by several closed surfaces. Using Carleman’s method, the asymptotic formulas for the distribution of eigenfunc- tions 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 asymp- totic 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 formu- las for the distribution of eigenfunctions of the boundary value oscillation problems are obtained for isotropic and anisotropic elastic bodies using in- tegral 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 three- dimensional 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.

1991Mathematics 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

1072-947X/99/0300-0107$15.00/0 c1999 Plenum Publishing Corporation

2. Throughout the paper we shall use the following notation: x = (x1, x2, x3),y= (y1, y2, y3) are points ofR³;|x−y|=€ P3

k=1(xk−yk)²1/2

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] withS0covering all otherSk while the latter surfaces not covering each other; Si∩Sk =∅fori6=k,i, k= 0, m; the finite domain bounded bySk

(k = 1, m) will be denoted byDk; D0 = D0∪€ ^m

k=0∪ Sk

, Dk = Dk∪Sk, k= 1, m.

Ifuandv are the three-component real vectorsu= (u1, u2, u3) andv= (v1, v2, v3), thenuv=P3

i=1uiviis the scalar product of these vectors;|u|=

€ P3

i=1u²_i1/2

. The matrix product is obtained by multipliing a row by a column; the sign [·]^T denotes the operation of transposition; ifA=kAijk3×3

is a 3×3 matrix, then|A|² = P3

i,j=1A²_ij. Any vector u= (u1, u2, u3) is treated as a 3×1 one-column matrix: u=kuik3×1;Ak =kAjkk³j=1 is the k-th column vector of the matrixA.

The vectoru= (u1, u2, u3) will be called regular inDk if ui∈C¹(Dk)∩C²(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 divu+ρω²u= 0, (1) whereu(x) = (u1, u2, u3) is the displacement vector, ∆ is the three-dimen- sional Laplace operator, ρ = const > 0 is the medium density, ω is the oscillation frequency,λandµare the elastic Lam´e constants satisfying the natural conditions

µ >0, 3λ+ 2µ >0.

We introduce the matrix differential operator A(∂x) =A_ij(∂x)

3×3, Aij(∂x) =δijµρ⁻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+ω²u= 0. (2)

The matrix-differential operator

T(∂x, n(x)) =T_ij(∂x, n(x))

3×3, where

Tij(∂x, n(x)) =λρ⁻¹ni(x) ∂

∂xj

+λρ⁻¹nj(x) ∂

∂xi

+µρ⁻¹δij

∂

∂n(x),

n(x) is an arbitrary unit vector at the point x(if x∈ Sk, thenk = 0, m is the normal unit vector external with respect to the domainD0) is called the stress operator.

It will be assumed that the domainsDk,k= 0, m0 are filled with homo- geneous isotropic elastic media with the Lam´e constantsλk,µk and density ρk, while the other domainsDk,k=m0+ 1, mare hollow inclusions. When the operatorsAandT containλk andµk instead ofλandµ, we shall write

k

AandT^k, respectively.

We introduce the notation u⁺(z) = lim

D03x→z∈Sk

u(x), k= 0, m; u⁻(z) = lim

Dk3x→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, ω²) =Γkj(x−y, ω²)

3×3, where

Γkj(x−y, ω²) =ρδkj

4πµ e^ik2r

r − 1

4πω²

∂²

∂xk ∂xj

e^ik1r−e^ik2r

r , (3)

iis the imaginary unit,r=|x−y|,k1 andk2 are the nonnegative numbers defined by the equalities

k₁²= ρω²

λ+ 2µ, k₂²=ρω²

µ . (4)

Let κ⁰ be an arbitrary real fixed positive integer and κ > κ⁰ be an arbitrary number. If in (3) we replaceω=iκ, then we obtain

Γkj(x−y,−κ²) =ρδkj

4πµ e^−κrc2

r + 1

4πκ²

∂²

∂xk∂xj

e^−κrc1 −e^−κrc2

r , (5)

wherec²₁= (λ+ 2µ)ρ⁻¹,c²₂=µρ⁻¹. Sinceµ >0 and 3λ+ 2µ >0, we have λ+µ >0 andλ+ 2µ > µ. Hence c1> c2 and c⁻₁¹ < c⁻₂¹. Letδ < _2c¹

1 be an arbitrary positive integer. Then (5) can be rewritten as

Γkj(x−y,−κ²) = ρδkj

4πµe^−ακre^−δ1κr

r +

+ 1 4πκ²

∂²

∂xk∂xje^−ακre^−δκr

r −e^−δ1κr r

‘

, (6)

whereδ1=_c¹₂ −α= _c¹₂ −c¹1 +δ >0.

Moreover,

(1) ∂

∂xj



e^−ακre^−δ1κr r

‘

= e^−ακr

r² e^−δ1κr

−1− 1

c2κr‘ ∂r

∂xj

; (2) ∂²

∂xk∂xj

e^−ακre^−δ1κr r

‘= e^−ακr

r³ e^−δ1κr 3 + 3

c2κr+

+1

c²₂κ²r²‘ ∂r

∂xj

∂r

∂xk

+e^−ακr

r³ e^−δ1κr

−1− 1 c2κr‘

δkj ∂r

∂xk

,

(3) the functions (κr)ⁿe^−δ1κr,n= 0,1,2, . . . are bounded in the inter- valκ∈[0,+∞).

Taking into account the above arguments, from (6) we obtain the esti- mates

ŒŒ

Œ∂ⁿΓpq(x−y,−κ²)

∂xⁱ₁∂x^j₂∂x^k₃

ŒŒ

β const

rⁿ⁺¹e^−ακr, (7) 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 pointy to the boundary ofDk.

Denoteρy(x) = max{r, ly}.and introduce an auxiliary matrix b^k

Γ(x−y,−κ²) =

” 1−

1− r^m ρ^m_y (x)

‘n•_k

Γ(x−y,−κ²), (8) where Γ(x^k −y,−κ²) is the Kupradze matrix of fundamental solutions for the operatorA(∂x)^k −κ²I (I is the 3×3 unit matrix).

We denote by B(y, ly) a sphere of radius ly and center at the point y, and by C(y, ly) its boundary. It is easy to verify that €

1 − ρ^m_y^rm(x)

n

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).

Forx∈Dk\B(y, ly) we have 1−

1− r^m ρ^m_y(x)

‘n

= 1 and

B(y,ly)3xlim→z∈C(y,ly)

” 1−

1− r^m ρ^m_y (x)

‘n•

= 1.

Thus

b^k

Γ(x−y,−κ²) =Γ(x^k −y,−κ²)

for x∈ Dk\B(y, ly), while, in passing the boundaryC(y, ly), the function b

k

Γ and its derivatives up to the (n−1)th order inclusive remain continuous.

We writeb^k Γ as b^k

Γ(x−y,−κ²) =Γ(x^k −y,−κ²)€

nr^m/ρ^m_y(x) +· · · .

It is easy ti find that b^k

Γ and its derivatives up to the (m−2)-th order inclusive are continuous forx=y, while for x∈B(y, ly) we have, by virtue of (7), the estimates

ŒŒ

ζ^sb^k

Γpq(x−y,−κ²)

∂xⁱ₁∂x^j₂∂x^k₃

ŒŒ

Œ≤conste^−ακr

l_y^m r^m−s−1, (9) p, q= 1,3, i+j+k=s; m≥s+ 1.

5. We determine the limit

xlim→y

hk

Γ(x−y,−κ²)−Γ(x^k −y,−κ0²)i

, x, y∈Dk, k= 0, m0. Taking into consideration the expansion

e^−κc1r r = 1

r − κ c1

+ κ²

2!c²₁r− κ³

3!c³₁r²+· · · , we obtain

(1) e^−κc2r

r −e^−κ0 c2r

r = κ⁰−κ c2

+κ²−κ²0

2!c²₂ r+· · ·, (2) e^−κc1r

r −e^−κc2r r =κ

− 1 c1

+ 1 c2

‘

+겐 1

2!c²₁ − 1 2!c²₂

‘ r+

+곐

− 1

3!c³₁+ 1 3!c³₂

‘r²+· · ·,

(3) ∂²

∂xk∂xj

e^−κc1r −e^−κc2r

r =겐 1

2!c²₁ − 1 2!c²₂

‘ ∂²r

∂xk∂xj

+ +곐 1

6c³₂ − 1 6c³₁

‘ ∂²r²

∂xk ∂xj

+· · · , (4) ∂²r²

∂xk∂xj = 2δkj.

By virtue of the above relations we have

xlim→y

hk

Γpq(x−y,−κ²)−Γ^kpq(x−y,−κ0²)i

=

= (κ⁰−κ)ρ^3/2_k

4πµ^3/2_k δpq+(κ⁰−κ)ρ^3/2_k 12π

” 1

(λk+ 2µk)^3/2− 1 µ^3/2_k

• δpq=

= (κ⁰−κ)ρ^3/2_k 12π

” 1

(λk+ 2µk)^3/2 + 2 µ^3/2_k

•

δpq, 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 problem of the operatorA(∂x)^k −κ0²I” to the 3×3 matrixG(x, y,−κ0²) = G(x, y,k −κ²0),x∈Dk,y∈D €

D= ^m∪⁰

k=0Dk

,x6=y,k∈0, m0 which satisfies the following conditions:

(1)∀x∈Dk,∀y∈D,x6=y:

k

A(∂x)G(x, y,^k −κ0²)−κ²0 k

G(x, y,−κ²0) = 0,k= 0, m0, (2)∀z∈Sk, ∀y∈D:G^◦+(x, y,−κ0²) =G^k −(x, y,−κ0²),

€^◦

T(∂z, n(z))G(z, y,^◦ −κ0²)+

=€^k

T(∂z, n(z))G(z, y,^k −κ²0)₋

,k= 1, m0, (3)∀z∈Sk, ∀y∈D:G^◦+(z, y,−κ0²) = 0, k= 0, m0+ 1, . . . , m;

(4)G(x, y,^k −κ²0) =Γ(x^k −y,−κ0²)−g(x, y,^k −κ0²),x∈Dk,y∈D,k= 0, m0, whereg(x, y,^k −κ0²) is a regular in Dk solution of the following problem:

(1)∀x∈Dk,∀y∈D:A(∂x)^k g(x, y,^k −κ0²)−κ0²

kg(x, y,−κ0²) = 0,k= 0, m0; (2)∀z∈Sk, ∀y∈D:g^◦+(z, y,−κ0²)−g^k−(z, y,−κ0²) =

=Γ(z^◦ −y,−κ²0)−Γ(z^k −y,−κ0²),

€^◦

T(∂z, n(z))g(z, y,^◦ −κ0²)+

−€^k

T(∂z, n(z))g(z, y,^k −κ²0)−

=

=T^◦(∂z, n(z))Γ(z^◦ −y,−κ0²)−T(∂^k z, n(z))Γ(z^k −y,−κ²0),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,−κ²0) is proved. As is known [8],G(x, y,−κ0²) possesses a symmetry property of the form

G(x, y,−κ²0) =G^T(y, x,−κ0²). (11) Moreover, we have the estimates [9]

∀(x, y)∈Dk×Dk :Gpq(x, y,−κ0²) =O(|x−y|⁻¹),

∂

∂xj

Gpq(x, y,−κ0²) =O(|x−y|⁻²), m, n, j= 1,3; k= 0, m0.





(12)

7. Let u(x) = u(x) and^k v(x) = v(x),^k x ∈ Dk, be arbitrary (regular) vectors of the classC¹(Dk)∩C²(Dk),k= 0, m0. Then the following Green formula is valid [8]:

m0

X

k=0

Z

Dk

(v^kA^ku^k+E(^k v,^k u))dx^k = Z

S

v◦⁺(T^◦u)^{◦ +}ds+

+

m0

X

k=1

Z

Sk

€◦

v⁺(T^◦u)^{◦ +}−^kv⁻(T^ku)^{k −}

ds, (13)

whereS=S0 m∪

k=m0+1Sk,

k

E(v,^k u) =^k ρ⁻_k¹(3λk+ 2µk)

3 divv^kdivu^k+ρ⁻_k¹µk

2 X

p6=q

∂v^kp

∂xq

+

+∂v^kq

∂xp

‘∂u^kp

∂xq +∂u^kq

∂xp

‘

+ρ⁻_k¹µk

3 X

p,q

∂v^kp

∂xp −∂v^kq

∂xq

‘∂u^kp

∂xp −∂u^kq

∂xq

‘ . (14)

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

For the regular inDk,k= 0, m0vectoru(x) the following general integral representation is valid [8]:

∀y∈Dk: uj(y) =−

m0

X

k=0

Z

Dk

k

Γj(x−y,−κ²)(A(∂x)^k u(x)^k −κ²u(x))dx^k +

+ Z

S

‚^k

Γj(z−y,−κ²)(T(∂^◦ z, n(z))u(z))^{◦ +}−

−u^◦+(z)T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)ƒ dzS+ +

m0

X

k=1

Z

Sk

‚^◦

Γj(z−y,−κ²)(T(∂^◦ z, n(z))u(z))^{◦ +}−

−Γ^kj(z−y,−κ²)(T^k(∂z, n(z))u(z))^{k −}ƒ dzS−

−

m0

X

k=1

Z

Sk

‚◦

u⁺(z)(T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)−

−u^k−(z)(T^k(∂z, n(z))Γ^kj(z−y,−κ²)ƒ

dzS, 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 tensorg(x, y,−κ²) asκ→

∞. To this end we consider the functional L[u] =

m0

X

k=0

Z

Dk

€^k

E(u,^k u) +^k κ²u^k2 dx−2

m0

X

k=1

Z

Sk

‚◦

u⁺(z)T^◦(∂z, n(z))×

×Γ^◦j(z−y,−κ²)−u^k−(z)(T^k(∂z, n(z))Γ^kj(z−y,−κ²)ƒ

dzS, (16) wherej = 1,2,3 is the fixed number andyis an arbitrary fixed point inDk, k= 0, m0, which is defined in the class of regular inDk, k= 0, m0, vector functions satisfying the conditions:

(1)∀z∈Sk : u^◦+(z)−u^k−(z) =Γ^◦j(z−y,−κ²)−Γ^kj(z−y,−κ²), (T^◦u(z))^{◦ +}−(T^ku(z))^{k −}=T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)−

Tk(∂z, n(z))Γ^kj(z−y,−κ²),k= 1, m0,

(2)∀z∈Sk : u^◦+(z) =Γ^◦j(z−y,−κ²), k= 0, m0+ 1, . . . , m.

Theorem 1.The functionalLtakes a minimal value foru=gj(x, y,−κ²).

Proof. Let ube an arbitrary vector from the domain of definition of the functionalL, and letv=u−gj. Then, with (14) taken into account, (16) implies

L[u] =L[v+gj] =

m0

X

k=0

Z

Dk

€^k

E(v^k+g^k_j,^kv+^kg_j) +κ²(v^k+g^k_j)²ƒ dx−

−2

m0

X

k=1

Z

Sk

‚(v^◦+g^◦_j)T^◦Γ^◦j−(^kv⁻+^kg_j⁻)T^kΓ^kjƒ ds=

=

m0

X

k=0

Z

Dk

‚^k

E(v,^{k k}v) + 2E(^k v,^k g^k_j) +E(^k g^k_j,^kg_j) +κ²(^kv²+ 2v^kkg_j+g^k_j²)ƒ dx−

−

m0

X

k=1

Z

Sk

‚◦

v⁺T^◦Γ^◦j−^kv⁻T^kΓ^kj

ƒds−2

m0

X

k=1

Z

Sk

€◦

g_j⁺T^◦Γ^◦j−g^k_j⁻T^kΓ^kj

ƒds=

=L[gj] +

m0

X

k=0

Z

Dk

‚^k

E(^kv,v) +^k κ²v^k2ƒ dx+ 2

m0

X

k=0

Z

Dk

‚^k

E(^kv,g^k_j) +

+κ^2kv^kg_jƒ dx−2

m0

X

k=1

Z

Sk

‚◦

v⁺T^◦Γ^◦j−^kv⁻T^kΓ^kj

ƒds. (17)

Using the Green formula (13) for v = u−gj, u = gj and taking into account that A^kkg_j =κ^2kg_j, ^◦v⁺(z) = 0 for z∈Sk,k = 0, m0+ 1, . . . , m, we obtain

m0

X

k=0

Z

Dk

‚^k

E(v,^{k k}g_j) +κ²v^kkg_jƒ dx=

m0

X

k=1

Z

Sk

‚_◦

v⁺(T^◦◦g_j)⁺−v^k−(T^kkg_j)⁻ƒ

ds. (18)

Now, since v^◦+(z) = ^kv⁻(z), (T^◦◦g_j)⁺−(T^kg^k_j)⁻ =T^◦Γ^◦j −T^kΓ^kj forz ∈Sk, k= 1, m0 (17) by virtue of (18) takes the form

L[u] =L[gj] +

m0

X

k=0

Z

Dk

‚^k

E(^kv,v) +^k κ²v^k2ƒ

dx≥L[gj].

Theorem 2. The estimate

ŒŒgjj(y, y,−κ²)−gjj(y, y,−κ0²)ŒŒ≤const ly^1+δ

, y∈D, δ >0, (19) holds for the function gjj(y, y,−κ²).

Proof. We write formula (15) foruj(x) =gjj(x, y,−κ²) and Γj(x−y,−κ²) = Gj(x−y,−κ²). Then taking into account the boundary and contact con- ditions forg andG, we get

∀(x, y)∈Dk:gjj(x, y,−κ²) =

=− Z

S

Γ◦j(z−x,−κ²)(T(∂^◦ z, n(z))G^◦j(z, y,−κ²))⁺dzS−

+

m0

X

k=1

Z

Sk

‚^◦

Gj+(z, x,−κ²)T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)−

−G^kj−(z, x,−κ²)T^k(∂z, n(z))Γ^kj(z−y,−κ²)ƒ dzS−

−

m0

X

k=1

Z

Sk

‚^◦

Γj(z−x,−κ²)(T^◦(∂z, n(z))G^◦j(z, y,−κ²))⁺−

−Γ^kj(z−x,−κ²)(T^k(∂z, n(z))G^kj(z, y,−κ²))⁻ƒ

dzS. (20)

Using formula (13) foru=v, (16) can be rewritten as L[u] =−

m0

X

k=0

Z

Dk

uk‚^k

A(∂x)u(x)^k −κ²u(x)^k ƒ dx+

Z

S

u◦⁺(z)(T^◦u(z))^{◦ +}dzS+

+

m0

X

k=1

Z

Sk

‚◦

u⁺(z)(T^◦u(z))^{◦ +}−u^k−(z)(T^ku(z))^{k −}ƒ dzS−

−2

m0

X

k=1

Z

Sk

‚◦

u⁺(z)T(∂^◦ z, n(z))Γ^◦j(z−y,−κ²)−

−u^k−(z)T^k(∂z, n(z))Γ^kj(z−y,−κ²)ƒ

dzS. (21)

which foru(x) =gj(x, y,−κ²) = Γj(x−y,−κ²)−Gj(x, y,−κ²) implies L[gj] =

Z

S

Γ◦j(z−y,−κ²)T(∂^◦ z, n(z))Γ^◦j(z−y,−κ²)dzS−

− Z

S

Γ◦j(z−y,−κ²)€^◦

T(∂z, n(z))G^◦j(z, y,−κ²)+

dzS+

+

m0

X

k=1

Z

Sk

‚^◦

Γj(z−y,−κ²)T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)−

−Γ^kj(z−y,−κ²)T(∂^k z, n(z))Γ^kj(z−y,−κ²)ƒ dzS+ +

m0

X

k=1

Z

Sk

‚^◦

Gj+(z, y,−κ²)T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)−

−G^kj−(z, y,−κ²)T^k(∂z, n(z))Γ^kj(z−y,−κ²)ƒ dzS−

−

m0

X

k=1

Z

Sk

‚^◦

Γj(z−y,−κ²)(T^◦(∂z, n(z))G^◦j(z, y,−κ²))⁺−

−Γ^kj(z−y,−κ²)(T(∂^k z, n(z))G^kj(z, y,−κ²))⁻ƒ

dzS. (22)

On the basis of (22), from (20) we obtain gjj(y, y,−κ²) =L[gj]−

Z

S

Γ◦j(z−y,−κ²)T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)dzS+

+

m0

X

k=1

Z

Sk

‚^◦

Γj(z−y,−κ²)T^◦(∂z, n(z))Γ^◦j(z−y,−κ²)−

−Γ^kj(z−y,−κ²)T^k(∂z, n(z))Γ^kj(z−y,−κ²)ƒ

dzS. (23)

The vectorb^k

Γj(x−y,−κ²) defined by (8) belongs to the domain of defi- nition of the functionalLand, sincegj(x, y,−κ²) imparts a minimal value to the functionalL, it is obvious that

L[gj]≤L[bΓj].

Now (23) implies

gjj(y, y,−κ²)≤L[bΓj]− Z

S

Γ◦j

T◦Γ^◦jds+

+

m0

X

k=1

Z

Sk

(Γ^◦j

T◦Γ^◦j−Γ^kj k

TΓ^kj)ds, y∈Dk. (24)

By virtue of the properties ofΓ, from (21) we obtainb L[bΓj] =−

Z

B(y,ly)

Γbj(AbΓj−κ²bΓj)dx+ Z

S

Γ◦j

T◦Γ^◦jds−

−

m0

X

k=1

Z

Sk

(Γ^◦j

T◦Γ^◦j−Γ^kj k

TΓ^kj)ds. (25)

By (25) and (24) we have gjj(y, y,−κ²)≤ −

Z

B(y,ly)

Γbj(AbΓj−κ²Γbj)dx, y∈Dk, k= 0, m0. (26)

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

ŒŒbΓmj(x, y,−κ²)ŒŒ≤ const ly

for s= 0,

ŒŒκ²bΓij(x, y,−κ²)ŒŒ≤κ²conste^−ακr l_y⁵ r⁴=

= const

l_y⁵ r²(κ²r²)e^−ακr≤const

l³_y for s= 0,

ŒŒAbΓj(x, y,−κ²)ŒŒ≤const

l³_y for s= 2.

Hence (26) implies

gjj(y, y,−κ²)≤ const l_y⁴ ·4

3πl³_y≤ const

ly ≤const l^1+δy

, (27)

whereδ >0 is an arbitrary integer.

Let us estimate gjj(y, y,−κ²) from below. For this we introduce the notation:

M[u] =

m0

X

k=0

Z

Dk

‚^k

E(u,^k u) +^k κ²u^k2ƒ

dx, M0[u] =

m0

X

k=0

Z

Dk

‚^k

E(u,^k u) +^k κ0²

uk²ƒ dx,

N[u] =

m0

X

k=1

Z

Sk

‚◦

u⁺(z)T^◦Γ^◦j(z−y,−κ²)−u^k−(z)T^kΓ^kj(z−y,−κ²)ƒ dzS.

Sinceκ0²≤κ², we have

L[gj(x, y,−κ²)] = minL[u] = min(M[u]−2N[u])≥min(Mo[u]−2N[u]).

Let the vector functionϕ(x, y) impart a minimal value to the functional M0[u]−2N[u]. Then ϕ(x, y) is a regular inDk (k= 0, m0) solution of the following problem:

(1)∀x∈Dk,∀y∈D:A(∂x)^k ϕ(x, y)^k −κ0²

ϕ(x, y) = 0,k k= 0, m0;

(2)∀z∈Sk,∀y∈D:ϕ^◦+(z, y)−ϕ^k−(z, y) =Γ(z^◦ −y,−κ²)−Γ(z^k −y,−κ²), (T^◦ϕ(z, y))^{◦ +}−(T^kϕ(z, y))^{k −}=T^◦Γ(z^◦ −y,−κ²)−T^kΓ(z^k −y,−κ²);

(3)∀z∈Sk,∀y∈D:ϕ^◦+(z, y) =Γ(z^◦ −y,−κ²),k= 0, m0+ 1, . . . , m.

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

∀(x, y)∈Dk:ϕ(x, y) =− Z

S

Γ◦j(z−x,−κ²)T^◦G^◦j(z, y,−κ²0)dzs+

+

m0

X

k=1

Z

Sk

‚^◦

Gj(z, x,−κ0²)T^◦Γ^◦j(z, y,−κ²)−G^kj−(z, x,−κ0²)×

×T^kΓ^kj(z−y,−κ²)ƒ dzs−

m0

X

k=1

Z

Sk

‚^◦

Γj(z−x,−κ²)(T^◦G^◦j(z, y,−κ0²))⁺−

−Γ^kj(z−x,−κ²)(T^kG^kj(z, y,−κ²0))ƒ

dzs. (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 rxy

, k= 0, m0, rxy =|x−y|. (29) In that case

L[gj(x, y,−κ²)≥M0[ϕ]−2N[ϕ]≥ −2N[ϕ] =