THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E

(1)

THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E

ⁿ

I.E.NIYOZOV AND O.I.MAKHMUDOV

Department of Mechanics and Mathematics, Samarkand State University Boulevard 15, 103004, Samarkand, Uzbekistan..

ABSTRACT: In this paper, we consider the problem of analytical continuation of solutions to the system of equations of thermoelasticity in a bounded domain. That is, we make a detailed analysis of the Cauchy problem regarding the values of thermoelasticity in bounded regions and the associated values of their strains on a part of the boundary of this domain.

ABSTRAK: Di dalam kajian ini, kami menyelidiki masalah keselanjaran analitik bagi penyelesaian-penyelesaian terhadap sistem persamaan-persamaan termoelastik di dalam domain bersempadan berdasarkan nilai-nilainya dan nilai tegasannya bagi sebahagian daripada sempadan domain tersebut, iaitu kami mengkaji masalah Cauchy.

KEYWORDS: Cauchy problem; system theory of elasticity; elliptic system; ill-posed problem; Carleman matrix; regularization

1. INTRODUCTION

In this paper, we consider the problem of analytical continuation of the solution of the system equations of the thermoelasticity in spacious bounded domain from its values and values of its strains on part of the boundary of this domain, i.e., we study the Cauchy problem. Since, in many actual problems, either a part of the boundary is inaccessible for measurement of displacement and tensions or only some integral characteristics are available. Therefore, it is necessary to consider the problem of continuation for the solution of elasticity system of equations to the domain by values of the solutions and normal derivatives in the part of boundary of domain.

The system of equations of thermoelasticity is elliptic. Therefore, the Cauchy problem for this system is ill-posed. For ill-posed problems, one does not prove the existence theorem: the existence is assumed a priori. Moreover, the solution is assumed to belong to some given subset of the function space, usually a compact one [1]. The uniqueness of the solution follows from the general Holmgren theorem [2]. On establishing uniqueness in the article studio of ill-posed problems, one comes across important questions concerning the derivation of estimates of conditional stability and the construction of regularizing operators. Our aim is to construct an approximate solution using the Carleman function method.

Let

x

= (

x

₁

, ….., x

_n) and y = (

y

₁

, ….., y

_n) be points of the n-dimensional Euclidean space

E

ⁿ,

D

a bounded simply connected domain in

E

ⁿ, with piecewise-smooth boundary consisting of a piece ∑ of the plane

y

_n = 0 and a smooth surface

S

lying in the half-space yn > 0.

(2)

Suppose U(x)=(u₁(x),...,u_n(x),u_n₊₁(x)) is a vector function which satisfies the following system of equations of thermoelasticity in

D

[3]:

0,

= ) ( ) ,

( U x

B ∂

_x

ω

(1)

where ( , )= ( , ) ,

1) ( 1) ( + × +

∂

∂x Bkj x n n

B ω ω

and ( , )= ( ) ( ) , , =1,..., ,

2

2 k j n

x B x

j k j

k x

j

k ∂ ∂

+ ∂ + +

∆

∂

ω δ µ ρω λ µ

, 1,...,

= ,

= ) , (

1) ( 1)

(

k n

B x

n x

n k

+

∂

− ∂

∂ ω γ

, 1,...,

= ,

= ) ,

1)

(

j n

i x B

j x

j

n

∂

∂ ∂

+

ω ωη

,

= ) ,

1)(

( 1)

(

θ

ω

ⁱ

ω

B_n₊ _n₊ ∂_x ∆+

δ

ij is the Kronecker delta, ω is the frequency of oscillation and λ, µ, ρ, θ its coefficients which characterize the medium, satisfying the conditions

0.

>

0,

>

0,

>

0,

>

2 3 0,

> η

θ γ ρ µ λ

µ +

The system (1) may be written in the following manner:







+ +

∆

+

− +

+

∆

0,

=

0

= )

( ²

divu i i v v

u gradv

divu grad u

θ ωη ω

ρω γ

µ λ µ

where U(x)=(u(x),v(x)).

This system is elliptic, since, its characteristic matrix is

, 1 0

0 0

0 )

( )

(

0 )

( )

(

= )

( 2

2 1

= 2 2 1

2

1 2

1

= 2 1

Λ

Λ Λ

Λ

Λ Λ

n i

n

i

n i

n

i

ξ ξ µ α ξ

µ ξ µ λ ξ

ξ µ α

ξ ξ µ α ξ

ξ µ α ξ

µ ξ µ λ ξ

χ ⁺ ⁺ ⁺ ⁺

+ +

∑

and for arbitrary

ξ = ( ξ

₁

,..., ξ

_n

)

with real components satisfying the conditions

1,

2

=

1

= i n

i

∑ ξ

^{we have}

0. ) (

= )

( ξ µ

²

λ + µ ≠ χ

det

Statement of the problem. Find a regular solution

U

of the system (1) in the domain

D

by using its Cauchy data on the surface

S

:

(3)

, ),

(

= ) ( )) ( , ( ), (

= )

(y f y R y U y g y y S

U ∂_y ν ∈ (2)

where R(∂_y,ν(y)) is the stress operator, i.e.,

( , ( ))= ( , ( ), ) =

1) ( 1) (+ × +

∂

∂y y Rkj y y n n

R ν ν γ

… … … …

… … … … . .

0 0 … … … ,

, ) , (

= ) , (

=T y Tkj y nxn

T ∂ ν ∂ ν

, 1,...,

= , ), ) (

( ) (

= ) ,

( k j n

y y y

T y _k_j

k j j k y

kj

ν λ ν µ ν µ λ δ ν

∂ + ∂

∂ + + ∂

∂

∂ ∂

)) ( ),..., ( (

= )

(y

ν

₁ y

ν

_n y

ν

is the unit outward normal vector on ∂D at a point

y

, ),

, , (

= f₁ f_n₊₁

f Κ g=(g₁,Κ,g_n₊₁) are given continuous vector functions on

S

.

2. CONSTRUCTION OF THE CARLEMAN MATRIX AND APPROXIMATE SOLUTION FOR THE CAP TYPE DOMAIN

It is well known that any regular solution U(x) of the system (1) is specified by the formula

−

∂

−

∫

_∂ ⁽Ψ⁽ ^, ^){ ⁽ ^, ⁽ ⁾⁾ ⁽ ^)}

= ) (

2U x x y R _y y U y

D

ω ν

~( , )} ( )) , ,

)) ( ,

~(

{R ∂_y y Ψ y−x ^*U y ds_y x∈D

−

ν ω

⁽³⁾

where the symbol ^*− means the operation of transposition, Ψ is the matrix of the fundamental solutions for the system of equations of steady-state oscillations of thermoelasticity: given by

, )

, (

= ) ,

( Ψ (+1)×(+1)

Ψ x ω kj xω n n

+













∂

− ∂

−

Ψ

∑

₊ ₊

j k l nl kj n j n

k l

j

k x x x

2 1)

( 1)

( 3

1

=[(1 )(1 ) 2

= ) ,

( δ α

πµ δ δ

δ ω

,

|

| )

|

| ( ]exp )

(1 )

(1 ₍ ₁₎ ₍ ₁₎ ₍ ₁₎ ₍ ₁₎ ₍ ₁₎

1)

( x

x i x

i x _k_n _j_n _l ^l

k n k n

j j n j n

k l

γ λ δ δ δ

γδ δ

ωηδ

β ₊ ₊ ₊ ₊ + ₊ ₊











∂

− ∂

∂ −

− ∂ +

0,

= 1,2,3;

= 2 ,

) )(

2 ( 2

) (

(1 1)

= (

3

1

= 2

3 2

1 2 2

2 1 1 1

l l l

l i l

πρω α δ λ

λ µ λ π

δ δ λ

α ωθ −

∑

− +

+

−

− ⁻ ⁻

0,

= 1,2,3;

= ), )(

2 ( 2

) (

1)

= (

3

1

= 2

1 2 2 2 1

l l l

l l

l l β

λ λ µ λ π

δ

β δ

∑

− +

+

−

, ) 2 (

= 0,

= 1,2,3;

= ) ,

( 2

) )(

( 1)

= (

₁² ² ¹

3

1

= 2

1 2 2

2 1 2 1 2

+

−

− +

−

− ∑ γ ρω λ µ

λ λ π

δ δ

γ λ k l k

l l l

l l

(4)

), , (

= ) ,

~ ( , )

,

~ (

= ) ,

~(

1) ( 1)

(

ω ω

ω

x x x

x Ψ_kj _n _n Ψ_kj Ψ_kj −

Ψ Π Π₊ _× ₊

= )) ( ,

~(

y R ∂_y

ν

… … … …

… … … … . .

0 0 … … … .

Definition. By the Carleman matrix of the problem (1),(2) we mean an (n+1)×(n+1) matrix Π(

y,x,ω, τ

) depending on the two points

y,x

and a positive numerical number parameter

τ

satisfying the following two conditions:

), , , ( ) , (

= ) , , , (

1) Π y x ω τ Ψ x − y ω + G y x τ

where the matrix G(y,x,

τ

) satisfies system (1) with respect to the variable y on D, and Ψ(y,x) is a matrix of the fundamental solutions of system (1);

), (

|) ) , , , ( ) , (

|

| ) , , , ( (|

2)

∫

_∂D\S Π y x

ω τ

+ R ∂y

ν

Π y x

ω τ

dsy ≤

ε τ

where

ε ( τ ) → 0,

^as τ →∞; here |Π| is the Euclidean norm of the matrix ,

||

=||Π ₍₊₁₎_×₍ ₊₁₎

Π _i_j _n _n i.e., _| _|= ²_.

1 2 1

1

= ,



 



 Π

Π

∑

ⁿ⁺ _i_j

j i

In particular, | |= ².

1 2 1

1

=



 





∑

⁺ m n

m

u U

From the definition of Carleman matrix it follows that.

Theorem 1. Any regular solution

U

(

x

) of system (1) in the domain

D

is specified by the formula

−

∂

∫

_∂

⁽ Π ⁽ ^, ^, ^, ^){ ⁽ ^, ⁾ ⁽ ^)}

= ) (

2 U x y x R

_y

U y

D

ω τ ν

~ ( , ) ( , , , )} ( )) , ,

{ R ∂

_y

Π y x

^*

U y ds

_y

x ∈ D

− ν ω τ

⁽⁴⁾

where Π(

y,x,ω, τ

) is the Carleman matrix.

Using this matrix, one can easily conclude the estimated stability of solution of the problem (1), (2) and also indicate effective method decision this problem as in [4 - 6].

With a view to construct an approximate solution of the problem (1), (2) we construct the following matrix:

, )

, , (

= ) , ,

( Π ( +1)×( +1)

Π y x

ω

kj y x

ω

n n ⁽⁵⁾

+













∂

− ∂

−

Π

∑

₊ ₊

j k l nl kj n

j n

k l

kj y x x x

2 1)

( 1)

( 3

1

=[(1 )(1 ) 2

= ) , ,

(

δ α

πµ δ δ

δ ω

), , , ( ] )

(1 )

(1 ₍ ₁₎ ₍ ₁₎ ₍ ₁₎ ₍ ₁₎ ₍ ₁₎

1)

( kn j n l l

k n k n

j j n j n

k

l y x k

x

i x + Φ











∂

− ∂

∂ −

− ∂

+

β ωηδ

₊

δ

₊

γδ

₊

δ

₊

δ

₊

δ

₊

γ

where

) , ) (

= ( ) , , ( )

( 2 2

2

0 u s

du ku x

y s u i

y s u i Im K k

x y x K C

n n

n n n

 +













− + +

+

Φ

∫

^∞ + ^ψ ⁽⁶⁾

(5)





≥ +

≥ 1, 1,

2

= , cos

1, ,

2

= ),

= ( )

( ⁰

m m

n ku

m m n ku ku uJ

ψ J

₀

( u )

-Bessel's function of order zero,

2 1 1 2

1

) ... ( )

(

= y − x + + y

_n₋

− x

_n₋

s

and





+

−

⋅

−

⋅

−

1.

2

= , 1)!

( 2) ( 2 1) (

2

= , 2)!

( 2) ( 2 1)

= ( , 2

2 =

m n m

n

m n m

C n C

n n

m

n n

m

n πω

π πω

K

(ω),ω =

u

+

iv

(

u

,

v

are real), is an entire function taking real values on the real axis and satisfying the conditions:

,

<

)

(u ≠∞for u ∞

K

K ( u ) = / 0,

. ,

0,...,

= ,

<

) , (

|=

) ( exp

|

sup ⁽ ⁾ ¹

1

R u m p

u p M K

Imk ^p

v

∈

∞

≥

ω ν

The following theorem was proved in [7].

Lemma 1. For function

Φ

(

y,x,k

) the following formula is valid

|,

=|

), , , ( ) (

= ) , ,

( y x k ikr g y x k r y x

C

_n

Φ ϕ

_n

+

_n

−

where

ϕ

_n - are fundamental solutions of the Helmholtz equation,

g

n(

y,x,k

) is a regular function that is defined for all

y

and

x

satisfies the Helmholtz equation:

0.

= )

(∂_y g_n −k²g_n

∆

In (6) we put

K ( ω ) = exp ( τω )

. Then

),

, (

= ) , ,

( y x k Φ y − x k

Φ

_τ

s u

du ku x

y s u i

x y s u Im i

k s x y C

n n

n n m

m

n  +











− + +

∂

− ∂

Φ −

∫

^∞

−

2 2

2 1 0

1 exp ( ) ( )

= ) ,

( τ ψ

τ

, ) sin (

) ( cos

) ( exp

= 2

2 2

2 1 0

1

du ku s

u s x u

y s u

x

y _m _n _n

m n

n τ ψ

α τ

τ 











+

− + + +

∂ −

− ∂ −

∫

^∞

− (7)

.

= ) ,

(

τ

τ ∂

Φ

− ∂ Φ′ y x k

, ) sin (

) (

exp

= ) ,

( 2

2 1 0

1

du ku s

u s u x s

y k

x y

C _m

m n n

n τ ψ

τ τ

+ +

∂

− ∂

−

Φ′ −

∫

^∞

−

), , ( )

( exp

= ) ,

( ₁

1

s s k

x y k

x y

C _m

m n n

n τ

τ ψ

τ′

∂

− ∂

−

Φ′ ₋

−











−

′

k k

s J

m n k s

k

>

), ) (

2 ( 1

2

= , ) (

cos

<

0,

=

2 2 0

2 2

τ τ

π τ

τ ψ_τ

(6)

Now, in formulas (5) and (6) we set

Φ ( y , x , k ) = Φ

_τ

( y − x , k )

and construct the matrix

Π ( y , x , ω ) = Π ( y , x , ω , τ )

From Lemma 1 we obtain,

Lemma 2. The matrix Π(

y,x,ω, τ

) given by (5) and (6) is Carleman's matrix for problem (1), (2).

Indeed by (5), (6) and Lemma 1 we have

), , , ( ) , , (

= ) , , ,

( y x ω τ Ψ y x ω + G y x τ Π

where

, ) , , (

= ) , ,

( y x τ G y x τ

G

_kj

+













∂

− ∂

−

− ₊ ₊

∑

j k l nl kj n

j n

k l

j

k y x x x

G

2 1)

( 1)

( 3

1

=[(1 )(1 ) 2

= ) , ,

(

δ α

πµ δ δ

δ τ

+













∂

− ∂

∂ −

− ∂

+ ₊ ₊ ₊ ₊

k n k n

j j n j n

k

l i

ωηδ

₍ ₁₎(1

δ

₍ ₁₎) x

γδ

₍ ₁₎(1

δ

₍ ₁₎) x

β

1.

1,...,

= , ), , , , (

1) ]

( 1)

( +

+

δ

_k_n₊

δ

_j_n₊

γ

_l g_n y x k_l

τ

k j n

By a straightforward calculation, we can verify that the matrix G(y,x,

τ

) satisfies the system (1) with respect to the variable

y

everywhere in

D

. By using (5), (6) and (7) we obtain

), ( exp ) (

|), ) , , , ( ) , (

|

| ) , , , (

(| ₁

\ n

m y

S y

D Π y x

ω τ

+ R ∂

ν

Π y x

ω τ

ds ≤C x

τ

−

τ

x

∫

_∂ ⁽⁸⁾

Where

C

1(

x

) is a bounded function inside of

D

. Let us set

. )) ( )}

, , , ( ) ,

~( { )}

( ) , ( ){

, , , ( (

= ) (

2 _y _y ^* _y

S y x R U y R n y x U y ds

x

U_τ

∫

Π

ω τ

∂

ν

− ∂ Π

ω τ

⁽⁹⁾

The following theorem holds.

Theorem 2. Let

U

(

x

) be a regular solution of the system (1) in

D

such that

.

\ ,

| ) ( ) , (

|

| ) (

| U y + R ∂

_y

ν U y ≤ M y ∈ ∂ D S

⁽10)

Then for τ ≥1 the following estimate is valid:

).

( exp ) (

| ) ( ) (

|U y −U_τ y ≤MC₂ xτ^m −τx_n By formulas (4) and (9) we have

. )) ( )}

, , , ( ) ,

~( { )}

( ) , ( ){

, , , ( (

|=

) ( ) (

|

2 ^*

\S y y y

D y x R U y R y x U y ds

x U x

U − _τ

∫

_∂ Π ω τ ∂ ν − ∂ ν Π ω τ

Now on the basis of (8) and (10) we obtain the required estimate.

Next we write out a result that allows us to calculate

U

(

x

) approximately if, instead of

U

(

y

) and R(∂_y,

ν

)U(y), its continuous approximations _f_δ₍_y₎ and g_δ(y) are given on the surface

S

:

(7)

1.

<

0 ,

| ) ( ) ( ) , ( max|

| ) ( ) (

max| f y − f_δ y + R ∂_y ν U y −g_δ y ≤δ δ

S S

(11) We define a function

U

_τ_δ

(x )

by setting

, )) ( )}

, , , ( ) ,

~( { ) ( ) , , , ( (

= ) (

2 _y ^* _y

S y x g y R y x f y ds

x

U_τ_δ

∫

Π

ω τ

_δ − ∂

ν

Π

ω τ

_δ ⁽12)

where

0.

>

max ,

= 1 ,

= ₀ ⁰ _n _n

n D n

x x M x

x ln

δ τ

Then the following theorem holds:

Theorem 3. Let

U

(

x

D

satisfying the condition (10). Then the following estimate is valid:

. ,

) (

| ) ( ) (

| ₃ ⁰ M x D

ln x

C x U x U

m xn

xn

 ∈



 



≤ 

− _τ_δ

δ δ

From all of the above results we immediately obtain a stability estimate.

Theorem 4. Let

U

(

x

D

satisfying the conditions:

S D y M y U R

y

U( )| | ( _y, ) ( )| , \

| + ∂

ν

≤ ∈∂

and

. 1,

<

0 ,

| ) ( ) , (

|

| ) (

| U y + R ∂

_y

ν U y ≤ δ δ y ∈ S

Then

, ) ( )

(

| ) (

| ₄ ⁰

m xn

xn

ln M x

C x

U 



 



≤  δ δ

where ds C

y C x

x

C _n _y

D

, ~

|

| 1

= ~ )

4(

∫

_∂ − is a constant depending on λ,

µ

, ω. Corollary 1. The limits

) (

= ) lim ( ), (

= ) lim (

0

x U x U x

U x

U _τ_δ

τ δ

τ→∞ →

hold uniformly on each compact subset of

D .

3.

REGULARIZATION OF SOLUTION OF THE PROBLEM (1), (2) FOR A CONE TYPE DOMAIN

Let

x = ( x

₁

, Κ , x

_n

)

and

y = ( y

₁

, Κ , y

_n

)

be points in Eⁿ, D_ρ be a bounded simply connected domain in

E

ⁿ whose boundary consists of a cone surface

1 >

0,

>

2 ,

= ,

=

:

₁ ₁² ₁² ²₁

ρ

ρ τ π

α τ

α

_ρ

y

_n

y + + y

_n₋ _ρ

tg y

_n

Σ Κ

and a smooth surface

S

lying in the cone. Assume x₀=(0,...0,x_n)∈D_ρ.

(8)

We construct Carleman matrix. In formula (5), (6) we set

( )

[ ]

^, ^>^0, ^>^1.

= )

(

ω

Eρ

τ ω

xn

τ ρ

K −

Then

0 >

), , (

= ) , ,

( y x k Φ y − x k k

Φ

_τ

s u

du ku x

y s u i

x y s u i Im E

k s x y C

n n

n n m

m

n  +











− + +

∂

− ∂

Φ −

∫

^∞

−

2 2

2 1 0

1 ( ( )) ( )

= ) ,

( _ρ τ ψ

τ (13)

.

= ) ,

( τ

τ

τ ∂

Φ

− ∂ Φ′ y x k

[ ]

{

⁽

}

⁽ ⁾ ^,

= ) ,

( 2

2 1 0

1

s u

du x ku

y s u i E s Im

k x y

C _m _n _n

m

n ′ + + − +

∂

− ∂

Φ′τ ⁻₋

∫

^∞ ρ

^τ ^ψ

Where

E

ρ(w) is Mittag-Löffer`s entire function [8]. For the functions Φ_τ(y−x,k) hold Lemma 1 and Lemma 2.

Further, we may show similar estimate for

U

_τ (

x

) and

U

_τδ (

x

) (in cone case) defined in (9) and (12), as Theorems 1, 2, 3, and 4.

For the simplicity let us consider

n

= 3, since the other cases are considered analogously.

Suppose that

D

_ρ is a bounded simple connected domain in

E

³ with boundary consisting of part ∑ of the surface of the cone

0,

>

1,

>

2 ,

= ,

=

₃² ₃

2 2 2

1

y y tg y

y ρ

ρ τ π

τ

_ρ _ρ

+

and of a smooth part of the surface

S

lay inside the cone. Assume

x

₀

= (0,0, x

₃

) ∈ D

_ρ. We construct Carleman`s matrix. In formulas (5), (6) we take

cos , )

( )

( 4

= 1 ) , ,

( 2

3 3 2

1

0 3 1

2 u s

du ku x

y s u i

w Im E

x E k

x

y + + − +

Φ

∫

^∞

ρ ρ

ρ ρ τ

τ τ

π

(14)

where

w = i u

²

+ s + y

₃. For the functions Φ_τ(y,x,k) holds Lemma 1.

It follows from the properties of

E

_ρ(w) that for

y ∈ Σ

and 0<u<∞ the function

) , , ( y x k

Φ

τ defined by (13) its gradient and the second partial derivatives 1,2,3,

= , ), , ,

2 (

j y k

y k x y

j k∂

∂ Φ

∂ _τ

tends to zero as

τ

→∞, for a fixed

x ∈ D

_ρ

.

Then from (5) we find that the matrix Π(

y,x,ω, τ

) and its stresses )

, , , ( ) ,

( ν y x ω τ

R ∂_y Π also converge to zero as

τ → ∞

for all

y ∈ Σ ,

i.e., Π(

y,x,ω, τ

) is the Carleman matrix for the domain

D

_ρ and the part

Σ

of the boundary.

(9)

If

U

(

x

) is a regular solution of the system (1) then the following integral formula . )) ( )}

, , , ( ) ,

~( { )}

( ) , ( ){

, , , ( (

= ) (

2 _y _y ^* _y

D y x R U y R y x U y ds

x

U

ω τ ν ν ω τ

ρ

Π

∂

−

∂

∫

_∂ Π

holds. For

x ∈ D

_ρ we denote by

U

_τ

(x )

the following:

. )) ( )}

, , , ( ) ,

~( { )}

( ) , ( ){

, , , ( (

= ) (

2 _y _y ^* _y

S y x R U y R y x U y ds

x

U_τ

∫

Π

ω τ

∂

ν

− ∂

ν

Π

ω τ

⁽¹⁵⁾

Then the following theorem holds.

Theorem 5. Let

U

(

x

D

_ρ such that

. ,

| ) ( ) , (

|

| ) (

| U y + R ∂

_y

ν U y ≤ M y ∈ Σ

⁽¹⁶⁾

Then for τ ≥1 the following estimate is valid:

), (

exp ) (

| ) ( ) (

| U x

₀

− U

_τ

x

₀

≤ MC

_ρ

x

₀

τ

³

− τ x

₃^ρ

where

x

₀

= (0,0, x

₃

) ∈ D

_ρ

, x

₃

> 0,

.

|,

=|

1 ,

= )

( ₃ ₀ ₀

0

0 ds r y x C constant

C r x

Cρ ρ

∫

_Σ _y − ρ −

Let us take continuous approximations

f

_δ(

y

) and

g

_δ(

y

) of

U

(

y

) and R(∂_y,ν)U(y), respectively, i. e.,

1

<

0 ,

| ) ( ) ( ) , ( max|

| ) ( ) (

max|U y − f_δ y + R ∂_y ν U y −g_δ y ≤δ δ

S S

and define the following function

, )) ( )}

, , , ( ) ,

~ ( { ) ( ) , , , ( (

= ) (

2

_y ^* _y

S

y x g y R y x f y ds

x

U

_τ_δ

∫ Π ω τ

_δ

− ∂ ν Π ω τ

_δ

Then the following theorem holds.

Theorem 6. Let

U

(

x

) be a regular solution of the system (1) in the domain

D

_ρ satisfying the condition (16), then

, ) ( ) (

| ) ( ) (

| ₀ ₀ ₀ ³

δ δ

ρ δ

τ

lnM x

C x U x

U − ≤ ^q

where =( _ρ ) ^ρ , ^ρ =max ( ₃)^ρ,

τ δ

τ

M R Re i s y

ln R

S

− +

1 .

= 1 ) ( , ) (

= ₄

0 3 0 0

3

dsy

r C r

x R C

q x 



 



 +

∫

_Σ

ρ ρ

ρ

The theorem is proved analogously as Theorems 3 and 4.

Corollary 2. The limits

) (

= ) lim ( ), (

= ) lim (

0

x U x U x

U x

U _τ_δ

τ δ

τ→∞ →

hold uniformly on each compact subset of

D

_ρ

.

(10)

REFERENCES

[1] Lavrent'ev, M. M. “Some Ill-Posed Problems of Mathematical Physics.” , Springer- Verlag, (1967).

[2] Petrovskii, I. G. “Lectures on Partial Differential Equations [in Russian].”, Fizmatgiz Press, Moscow(1962).

[3] Kupradze, V. D., Burchuladze, T. V., Gegeliya, T. G. “Three-Dimensional Problems of the Mathematical Theory of Elasticity [in Russian]”, Nauka Press, Moscow(1968).

[4] Makhmudov, O. I., Niyozov, I. E. “The Cauchy problem for the Lame system in infinite domains in Rm. Journal of inverse and Ill-Posed Problems, 14(2006):905-24.

[5] Makhmudov, O. I., Niyozov, I. E. “Regularization of a solution to the Cauchy Problem for the System of Thermoelasticity.” Contemporary Mathematics. AMS, Primary, 382(2005):35-72.

[6] Makhmudov, O. I., Niyozov, I. E., Tarkhanov, N. “The Cauchy Problem of Couple-Stress Elasticity.” Contemporary Mathematics. AMS, 455(2008):207-310.

[7] Yarmukhamedov, Sh. Ya. “Cauchy problem for the Laplace equation.” Journal.

Dokl.Acad.Nauk SSSR [Soviet Math.Dokl.], 357(1997):320-357.

[8] Dzharbashyan, M. M. “Integral Transformations and Representations of Functions in a Complex Domain [in Russian],”Nauka Press, Moscow. (1966)