1
UDC 517.946
THE CAUCHY PROBLEM FOR SYSTEMS OF ELLIPTIC TYPE
EQUATIONS OF THE FIRST ORDER IN m  DIMENSIONAL
BOUNDED DOMAIN
D. A. Juraev
Karshi branch, TITU
In this work is considered the Cauchy problem for systems of elliptic type
equations of the first order with constant coefficients factorable Helmholtz
operator in m  dimensional bounded domain.
The problem in the task relates to the ill-posed problems, i.e. it is unstable.
If the narrow class of possible solutions to the compact, conditional stability of
problem follows from Tikhonov A.N. [4].
Using the ideas of M.M. Lavrentiev [2] Sh. Yarmukhamedov was built
explicitly regularized problem for the Laplace equation [3]. Many posed
problems for systems of elliptic type equations of the first order with constant
coefficients factorable operator Helmholtz available calculation value of the
vector function on the entire boundary. Therefore, the task of reconstruction,
solving systems of elliptic type equations of the first order with constant
coefficients factorable of Helmholtz operator, is one of the most urgent
problems of the theory of differential equations.
Let Rm (m  3) the real m  dimensional Euclidean space,
х  ( х1 , ..., xm )  R m , у  ( у1 , ..., ym )  R m , х  ( х1 ,..., хm 1 )  R m - 1 , у   ( у1 ,..., уm 1 )  R m - 1 .
G  R m  bounded simply connected domain whose boundary consists of a
compact part T of the plane ym  0 and smooth surface S ( S  C 1 ) , of the piece
lying on the half-space ym  0 . xT  ( х1 ,..., xm )T  transposed vector x ,
  y   x , r  y  x , s   2 , w  i u 2   2  уm , u  0, w0  i  ym ,
T
   
 
 , U ( x )  (U1 ( x ),U 2 ( x ), ... ,U n ( x )) T , u 0  (1,1, ... ,1)  R n ,
 
,
,...,
x  x1 x2
xm 
E (z )  diagonal matrix, z  ( z1 , z2 , ... , zn )  R n .
Let p( x)   p1 ( x), ... , pl ( x)T , where p1 ( x), . .. , pl ( x) polynomials х1 ,..., xm with
real coefficients, such that for p j ( x )  0 at x  R m \ 0, j  1, l . (This condition
 
 - elliptical [1]).
 x 
means that the operators p j 
We denote through Al n ( p( x), x) , (l , n  3) the class of matrices D( x T ) with
elements in the set of linear functions with complex coefficients, for which the
condition:
2
D  ( x T ) D( x T )  E (( x  2 )u 0 ) ,
Where D ( xT )  the conjugate to the matrix D( xT ) ,   a real number.
2
Consider in area G a system of differential equations
 
D U ( x)  0,
 x 
T
where D( x )  Al n ( p( x ), x ), (l , n  3)  the characteristic matrix.
(1)
We denote through H (G )  the class of vector-valued functions in the G
continuous on G  G  G and satisfy the system (1).
Cauchy problem. Let U ( y )  H (G ) and
(2)
U ( y ) S  f ( y ), y  S ,
Here, f ( y )  given continuous vector function on S .
Want to restore the vector function U ( y ) of the area G , of the values f ( y ) .
Denote through K ( w), w    i  the entire function that takes real values
for real w (  ,  real numbers) satisfying the conditions:
K ( )  0, sup  p K p ( w)  ( , p)  ,      , p  0,1, ..., m, m  3 ,
(3)
 1
Define a function  ( y, x ) at y  x in the following equality:

 k 1
K ( w) cos u
Cm K ( xm ) ( y , x )  k 1  Im
du, m  2k  1, k  1,
s 0
w  xm u 2   2
(4)
Cm  ( 1) k 2 2 k 1 ( k  1)! ( m  2)m ,
Where m  the area to surfaces of the single ball in space Rm .
If U ( y )  H (G ) , then we have the following Cauchy type integral formula
[1]
U ( x )   M ( y, x )U ( y )ds y , x  G ,
(5)
G
where
Here
m2
 


 i   2
(1)
0    
T
M ( y, x )   E  
 H m  2 (  u )u  D    D ( t ) ,
4
4

r

y

  
  
2

 

t  (t1 , ..., tm )  the unit outward normal at points y , surfaces
i  


4  4r 
H
(1)
m2
2
m2
2
G ,
H m(1) 2 (u )  the fundamental solution of the Helmholtz equation in Rm ,
2
(u )  the Hankel function of the first kind of order
m2
.[5]
2
In (4) choosing K ( w)  exp( w2 ),   0 , we

e x m  k 1
exp( w2 ) cos u
 ( y , x ) 
Im
du ,
Cm s k 1 0
w  xm
u2   2
2
i  
Equation (5) is correct, if we put 

4  4r 
m2
2
(6)
H m(1) 2 (u ) together a function
2
3
i  
 ( y , x )   

4  4r 
m2
2
H m(1) 2 (u )  g ( y, x ) ,
(7)
2
where g ( y, x)  the regular solution of the Helmholtz equation in the variable
y , including the point y  x .
Then the integral formula is:
U ( x )   N ( y, x )U ( y )ds y , x  G ,
(8)
G
where

  
N ( y, x )   E  ( y, x )u 0 D    D(t T ) .
 y  

Theorem. Let U ( y )  H (G ) satisfies inequality
U ( y )  1, y  T ,
(9)
If
U ( x )   N  ( y , x )U ( y )ds y , x  G,
(10)
S
then
U ( x )  U  ( x )  C ( x ) e  xm ,   1, x  G,
2
(11)
where C (x )  is a function that depends only on x .
Below for convenience, features, depending on x we denoted through
C (x ) . In the various inequalities they are different.
Proof. Using (8) and (10), we obtain
U ( x )   N ( y, x )U ( y )ds y  U ( x )   N ( y, x )U ( y )ds y , x  G .
G
T
Using (9), we estimate the following
U ( x )  U  ( x )   N  ( y, x )U ( y )ds y   N  ( y, x ) ds y , x  G ,
T
We estimate the integrals for this
  ( y, x ) ds , 
y
T
and

T
(12)
T
T
 ( y , x )
ds y , j  1, ..., m  1
y j

( y, x ) ds y parts T plane ym  0 .
ym
The imaginary part of (6), we obtain
e ( y m  x m )
 ( y, x ) 
2 2
2


0
e  ( u
2
 2 )
2
  e  ( u 2  2 ) cos 2ym u 2   2
cos u du 

u2  r2
 0
( ym  xm ) sin 2ym u 2   2
u2  r2
y  x, xm  0,
Using (13) we have

du ,
u 2   2 
cos u
(13)
4
  ( y, x ) ds
 C ( x )e x m ,   1, x  G ,
2
y
(14)
T
To estimate the second integrals we use the equality
 ( y, x )  ( y, x ) s
 ( y, x )

 2( y j  x j ) 
, j  1, ..., m  1.
y j
s
y j
s
(15)
Using (15), we obtain

T
2
 ( y , x )
ds y  C ( x ) e x m ,   1, x  G ,
y j
We estimate the integral

T

T
(16)

( y, x ) ds y , we obtain
ym
2

( y, x ) ds y  C ( x )e x m ,   1, x  G,
ym
(17)
From (14), (16) and (17) we obtain (11).
The theorem is proved.
Corollary. Limiting equality
lim U  ( x)  U ( x) ,
 
holds uniformly on each compact set of area G.
List of references
1. Tarkhanov N.N. Integral representation of solutions of linear differential
equations of the 1st order partial and some of its applications. // Some
problems of complex analysis, Institute of Physics of the USSR,
Krasnoyarsk, 1980, pp. 147-160.
2. Lavrentiev M.M. On the Cauchy problem for the Laplace equation. //
Notify Academies of the Sciences of USSR, Mathematical Collection,
Vol. 20, 1956, pp. 819-842.
3. Yarmukhamedov Sh. Carleman function and the Cauchy problem for the
Laplace equation. // Siberian Mathematical Journal, Vol. 45, No. 3, 2004,
pp. 702-719.
4. Tikhonov A.N. On the solution of ill-posed problems and the method of
regularization. // Reports of the Academy of Sciences of USSR, Vol. 151,
No. 3, 1963, pp. 501-504.
5. Changmei L. The Helmholtz equation on Lipschitz domains. //
Department of Mathematics. University of North Carolina. Chapel Hill,
NC 27599-2350 September, 1995, pp. 1-28.