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Об определении неизвестных коэффициентов в квазилинейном эллиптическом уравнении.

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!
∀# ∃ ∀
.. 1
. , . ! . ∀ #, .
: , .
1. ∃ [1, 2]. [3–10]. .
! D – n - E n , x = ( x1 , x2 ,..., xn ) –
D , – D , p0 , p1 – , Q ≡ [ p0 , p1 ] . ! I = {1, 2,..., n} , i0 ∈ I .
∀ {k1 (u ), k2 (u ),..., kn (u ), q (u ), u ( x, p )} #∃ :
n
− ki (u )u xi xi + q(u )u = h( x, p ) , x ∈ D , p ∈ Q ,
(1)
i =1
u ( x, p )
= f (ξ , p), ξ ∈ , p ∈ Q ,
ki ( Fi )uν (ξi , p ) = gi ( p ), ξi ∈ , i = 1, 2,...n, p ∈ Q,
ki0 ( Fn +1 )uν (ξ n +1 , p ) = q ( Fn +1 )φ ( p) + g n +1 ( p ), ξ n+1 ∈ , p ∈ Q ,
(2)
(3)
(4)
ξi , i = 1, 2,..., n + 1 – , Fi ≡ Fi ( p ) = f (ξi , p ), i = 1, 2,..., n, h( x, p),
φ ( p ), f (ξ , p), gi ( p ), i = 1, 2,..., n + 1 – , gi ( p) ∈ Lip (Q ), i = 1, 2,..., n + 1, φ ( p ) ∈
∈ Lip(Q), h( x, p), f (ξ , p ) # p ∈ Q C α ( D), C 2+α ( ),
0 < α < 1 , v – , ∂u
ξi , i = 1, 2,..., n + 1, uν (ξi , p) ≡ (ξi , p ), i = 1, 2,..., n + 1, Lip – , ∂v
#∃ # %, R1, R2 – .
&, (3) # # ξi , i = 1,2,..., n . (4) , ξ n+1 ∋#. ! 1
( ∀ ( – , , (
, . ).
e-mail: ramizaliyev3@rambler.ru
4
, 32, 2011
..
(1) q (u )u ( x, p ) . ki (u ), i = 1, 2,..., n , (3), , q (u ) , (4).
! ki (u ), i = 1, 2,..., n [3] .
1. ∗ {k1 (u ), k2 (u ),..., kn (u ), q (u ), u ( x, p )} (1)–
(4), 0 < ki (u ) ∈ C[ R1 , R2 ], i = 1, 2,..., n, 0 < q (u ) ∈ C [ R1 , R2 ] , u ( x, p) ∈ C ( DxQ ), u ( x, p) #-
p
C 2 ( D),
∃# u xi ( x, p), i = 1, 2,..., n
x → ξi ∈ ,
i = 1, 2,..., n + 1, R1 ≤ u ( x, p ) ≤ R2 # (1)–(4).
!
, f (ξ , p ) = f1 (ξ ) f 2 ( p ) . min f1 (ξ ) = r1 , max f 2 (ξ ) = r2 . !
ξ ∈
ξ ∈
, Fi ( p), i = 1, 2,..., n + 1 # # i ( Fi ), i = 1, 2,..., n + 1 [ r1 , r2 ] Q ∃ Lip (Q) . !
,
ki (u ) ∈ Lip[ R1 , R2 ], i = 1, 2,..., n q (u ) ∈ Lip[ R1 , R2 ] [ R1 , R2 ] / [ r1 , r2 ]
# , ∃ %.
∋
, (1)–(4) ∃, ki (u ) ∈ Lip[ R1 , R2 ], i = 1,2,..., n, q (u ) ∈ Lip[ R1 , R2 ],
u ( x, p) ∈ C 2+α ( D) ∀p ∈ Q u ( x, p) p # %. +,
∃ , u ( x, p) ∈W p1 2 ( D) ⊂ C1+α ( D) p1 > n, ∀p ∈ Q . ! (3) (4)
, ki (u ) ∈ Lip [ R1R2 ] , i = 1,2,..., n,
q (u ) ∈ Lip[ R1, R2 ] . ,
u ( x, p) ∈ C 2+α ( D)
∀p ∈ Q # % p [11] .
2. %
!, (1)–(4), ∃ (1) – (4) , , ∃ (1)–(4)
#∃ . ! Z ( x, p ) = u ( x, p ) − u ( x, p ), λi (u , u ) = ki (u ) − ki (u ), i = 1,2,..., n, µ (u , u ) = q (u ) − q(u ),
δ1 ( x, p ) = h( x, p) − h( x, p), δ 2 (ξ , p ) = f (ξ , p ) − f (ξ , p), δ 3 ( p ) = φ ( p ) − φ ( x), δ1i ( p ) = g i ( p) − gi ( p),
i = 1, 2,..., n + 1.
− δ 2 ( x, p ) # , #∃# δ 2 (ξ , p )
# p ∃# C 2+α ( D) . ∗ f ( x, p) f (ξ , p ) C 2+α ( D ).
2. ε > 0 δ = δ (ε ) > 0 , max δ1 ( x, p ) C ( D ) < δ , max δ2 ( x, p ) 2
< δ , max δ 3 ( p ) < δ , max δ1i ( p ) < δ , i = 1, 2,..., n + 1 (5)
p
p
C ( D)
p
p
# Z ( x, p ) < ε , λi (u , u ) < ε , i = 1,2,..., n, µ (u , u ) < ε x ∈ D,
p ∈Q ,
, (1)–(4) .
(1)–(4) ∃ 1.
& 1. ! g1 ( p) ≠ 0 , φ ( p ) ≠ 0 , N ⋅ mesD < 1 . ,
(1)–(4) . N – , ∃ .
+. . (1) − (4) (1)–(4) Z1 ( x, p) =
= Z ( x, p ) − δ ( x, p ) . ,
2
% «. &.
∋», ()∗+ 5
5
n
n
i =1
i =1
− k i (u ) Z1xi xi + q (u ) Z1 = δ 4 ( x, p ) + α i ( x, p)λi (u , u ) +β ( x, p) µ (u , u ),
(6)
Z1 ( x, p ) = 0 ,
(7)
λi ( Fi , Fi ) = δ 5 ( p ) + γ i ( p ) Z1ν (ξi , p ), i = 1, 2,..., n,
µ ( Fn +1 , Fn +1 ) = δ 6 ( p) + γ n +1 ( p ) Z1v (ξ n +1 , p) + γ n + 2 ( p )λi0 ( Fn +1 , Fn +1 ) ,
(8)
(9)
n
α i ( x, p) = u xi xi , i = 1, 2,..., n, β ( x, p ) = −u , δ 4 ( x, p ) = δ1 ( x, p) +
k i (u )δ2 x x ( x, p) − q(u )δ2 ( x, p),
i i
i =1
γ i ( p) = −ki ( F i )[uv (ξi , p)]−1 , i = 1, 2,..., n, δ 5 ( p) = [δ1i ( p) − k i ( F 1 )δ2v (ξi , p)] × [uν (ξi , p )]−1 , i = 1, 2,..., n,
δ ( p) = [−δ
( p ) − q ( F n+1 )δ ( p ) + k i ( F )δ (ξ , p)] × [φ ( p )]−1 , γ ( p ) = −k ( F )[φ ( p )]−1 ,
6
1, n +1
3
0
2
2v
n +1
n +1
i0
n +1
−1
γ n + 2 ( p ) = uν (ξ n +1 , p )[φ ( p)] .
! ∃ / [12] (6)–(7) # Z1 ( x, p ) # (8) (9) . ,
n
∂
G (ξi ,θ ) {δ 4 (θ , p ) + α i (θ , p )λi (u , u ) + β (θ , p ) µ (u , u )}dθ , i = 1, 2,..., n,
∂v
i =1
D
λi ( Fi , Fi ) = δ 5 ( p ) + γ 1 ( p ) µ ( Fn +1 , Fn +1 ) = δ 6 ( p ) + γ n+1 ( p )
n
∂
G
(
ξ
,
θ
)
δ
(
θ
,
p
)
+
α i (θ , p)λi (u , u ) + β (θ , p ) µ (u , u ) }dθ +
{
1
4
∂v
i =1
D
+γ n + 2 ( p)λi0 ( Fn+1 , Fn+1 ).
(10)
n
, (10) χ = max λi (u , u ) + max µ (u , u ) .
u
i =1
u
,
(10) , λi ( Fi , Fi ) ≤ δ 7 ( p ) + χ K1 (ξi ,θ )dθ , i = 1, 2,..., n,
D
(11)
µ ( Fn+1 , Fn+1 ) ≤ δ 8 ( p) + χ K 2 (ξ n +1 ,θ )dθ + γ 2 ( p ) λi0 ( Fn +1 , Fn+1 ) .
D
&
δ i ( p ) , i = 7,8 – , ∃ # (5) .
∗ / K i (ξi ,θ ) , i = 1, 2 # #∃# [12] :
G ( x,θ ) ≤ M 1 x − θ
2−n
, M1 > 0, K i (ξi ,θ ) ≤ M i +1 ξi − θ
1− n
/
> 0, M i +1 > 0, i = 1,2,..., n.
(12)
! D (11) ∃# M [mesD]1/ n , M > 0 . . (11) χ ≤ δ 9 ( p ) + χ NmesD .
! # β = NmesD < 1 . ,
χ χ ≤ (1 − β )−1δ 9 ( p) .
0
, χ # δ → 0 (5) ., .
3. ∋
1
(1)–(4) #∃ :
n
− ki( S ) (u ( S ) )u x( Sx+1) + q ( S ) (u ( S ) )u ( S +1) = h( x, p) , x ∈ D , p ∈ Q ,
i =1
i i
u ( S +1) ( x, p ) = f (ξ , p ) , ξ ∈ , p ∈ Q ,
6
(13)
(14)
, 32, 2011
..
ki( S +1) (u ( S +1) )uv( S +1) (ξi , i (u ( S +1) )) = gi (i (u ( S +1) )) , ξi ∈ ,
ki( S +1) (u ( S +1) )uv( S +1) (ξ n +1 , n +1 (u ( S +1) ))
0
=q
( S +1)
(u
( S +1)
) φ (n+1 (u
i = 1, 2,..., n,
( S +1)
(15)
)) + g 2 (n +1 (u
( S +1)
)) , ξ n+1 ∈ , (16)
i ( Fi ) , i = 1, 2,..., n + 1 # Fi ( p) =
= f (ξi , p ), i = 1, 2,..., n + 1 . ! (13)–(16) #∃
: # ki(0) (u (0) ) > 0, i = 1, 2,..., n, q (0) (u (0) ) > 0, ∃
Lip[ R1 , R2 ] , # (13) . + (13)–(14) u (1) ( x, p) . ! uv(1) (ξi , Φ i (u (1) )),
i = 1, 2,..., n + 1 (15), (16) ki(1) (u (1) ), i = 1, 2,..., n q (1) (u (1) ) # #∃ .
& 2. ! (1)–(4) ∃ s = 0,1,...,
u ( S ) ( x, p) ∈ C ( DxQ), u ( S ) ( x, p)
#
p C 2 ( D),
ki( S ) (u ( S ) ) ∈ Lip [ R1 , R2 ] ,
i = 1, 2,..., n, q ( S ) (u ( S ) ) ∈ Lip[ R1 , R2 ], gi ( p)uv( S ) (ξi , p ) > 0, i = 1, 2,..., n,
φ ( p )uv( S ) (ξi , p ) > 0 ,
g n+1 ( p )uv( S ) (ξ n +1 , p ) < 0 , NmesD < 1 , u ( S ) ( x, p) x .
,
{k
(S )
1
(u ( S ) ), k2( S ) (u ( S ) ),..., kn( S ) (u ( S ) ),
q ( S ) (u ( S ) ), u ( S ) ( x, p )} , (13)–(16), s → +∞ # (1)–(4) # . N – , ∃ .
∀. ! Z ( S ) ( x, p ) = u ( x , p ) − u ( S ) ( x, p ) ,
λi( S ) (u , u ( S ) ) = ki (u ) − ki( S ) (u ( S ) ), i = 1, 2,..., n, µ ( S ) (u , u ( S ) ) = q (u ) − q ( S ) (u ( S ) ).
% , # n
n
− ki (u ) Z x( S x+1) + q (u ) Z ( S +1) = α i ( S ) ( x, p )λi( S ) (u , u ( S ) ) + β ( S ) ( x, p) µ ( S ) (u , u ( S ) ),
i =1
i i
(17)
i =1
Z ( S +1) ( x, p)
= 0,
(S )
( S +1)
λ
( Fi , Fi ) = γ 1 ( p ) Zν
(ξ1 , p ), i = 1,2,..., n,
µ ( S +1) ( Fn+1 , Fn+1( S +1) ) = γ n +1 ( p ) Zν( S +1) (ξ n +1 , p ) + γ n( S+)2 ( p)λ ( S +1) ( Fn+1 , Fn +1( S +1) ),
( S +1)
(S )
(18)
(19)
(20)
α i( S ) ( x, p ) = u x(iSx+i 1) , i = 1,2,..., n , β ( S ) ( x, p) = −u ( S +1) , γ i ( S ) ( p ) = −k i ( Fi )[u ν ( S +1) (ξi , p )]−1 ,
γ n +1 ( p ) = −ki0 ( Fn +1 )[φ ( p )]−1 , γ n( S+)2 ( p ) = uν( S +1) (ξn +1 , p )[φ ( p )]−1.
∗ / (17)–(18) Z ( S +1) ( x, p ) # (17) (19) (20). ,
λi( S +1) ( Fi , Fi ( S +1) ) = γ 1( S ) ( p) Gν (ξi ,θ ) {
D
n
αi ( S ) (θ , p)λ i( S ) (u, u ( S ) ) + β (s) (θ , p) µ (s) (u, u(s) ) } dθ ,
i =1
n
µ ( S +1) ( Fn+1 , Fn +1( S +1) ) = γ n +1 ( p) Gν (ξ 2 , θ ) α i ( S ) (θ , p)λ i ( S ) (u , u ( S ) ) + β ( s ) (θ , p ) µ ( s ) (u , u ( s) ) dθ +
i =1
D
+γ n( S+)2 ( p )λi( S +1) ( Fn +1 , F n+1 ( S +1) ) .
0
(21)
! n
χ ( S ) = max λi ( S ) (u , u ) + max µ ( S ) (u , u ) .
u
% «. &.
i =1
u
∋», ()∗+ 5
7
! (21) , χ ( S +1) ≤ χ ( S ) NmesD . , .
4. (# 0∃ (1)–(4) .
1. ! ki (u ) > 0, i = 1, 2,..., n – , (1)–(2), (4) {q (u ), u ( x, p )} .
& 3. ! h( x, p ) = 0, f (ξ , p ) ≥ 0, g 2 ( p) = 0, φ ( p ) < 0 , NmesD < 1 . ,
(1)–(2), (4)
. N – , ∃ .
∀. ∋
, q (u ) . [11] (12)–(13) u ( S +1)
C (D)
≤ f
#
C( )
{
}
{q
, .. u ( S ) ( x, p) . + -
(S )
}
(u ( S ) ) .
!
q ( S ) (u ( S ) )u ( S +1) # (12) ∃# / u ( S +1) ( x, p ) :
n
u ( S +1) ( x, p ) = G ( x,θ ) ki (u ( S ) ) fθiθi (θ , p ) − q ( S ) (u ( S ) )u ( S +1) dθ + f ( x, p) .
i =1
D
!
(16) x = ξ n+1 , q ( S +1) ( Fn+1 ) = F ( S ) ( p ) − K ( S ) (ξ n +1 ,θ )q ( S ) (u ( S +1) )u ( S +1) (θ , p )dθ ,
(22)
D
n
F ( S ) ( p ) = ki0 ( Fn +1 )[φ ( p)]−1 fν (ξ n+1 , p) + Gv (ξ n+1 ,θ ) ki (u ( S ) ) fθiθi (θ , p ) dθ ,
i =1
D
−1
K ( S ) (ξ n +1 ,θ ) = [φ ( p )] ki0 ( Fn +1 )Gv (ξ n+1 ,θ ) .
+ K ( S ) (ξ n +1 ,θ ) (12). ! q1( S ) = ma q ( S ) (u ( S ) ) , R1 ≤ u ( S ) ≤ R2 .
,
(22) D q1( S +1) ≤ N1 + NmesDq1( S ) .
&
N1 > 0 s . β = NmesD . 2, 0 < β < 1 , 1
q1( S +1) ≤
N1 + q1(0) .
1− β
,
{q
(S )
}
(u ( S ) ) . , , q ( S ) (u ( S ) ) – -
. ,
∃ , {u
(S )
}
( x, p )
W p21 ( D ),
p1 > n, ∀p ∈ Q . ! u ( S ) ( x, p) C1 ( D ). ! (16) , -
{
}
q ( S ) (u ( S ) ) C[ R1 , R2 ] . #
(13)–(14) -
{
}
u ( S ) ( x, p) C 2 ( D ). (13)–(14), (16) s → +∞ , ∃ {q (u ), u ( x, p )} , #∃ (1)–(2), (4).
, .
8
, 32, 2011
..
2. ! q (u ) – . y = x2 (1)–(3) n = 2 {k1 (u ), k2 (u ), u ( x, y, p)} . {k1 (u ), k2 (u ), u ( x, y )} :
− k1 (u )u xx − k2 (u )u yy + q (u )u = h( x, y ), ( x, y ) ∈ D,
u ( x,0 ) = ϕ1 ( x), u(x, l2 ) = ϕ 2 ( x) ,
u(0,y ) = φ1 ( y ), u (l1 , y ) = φ2 ( y ) ,
k1 (φ1 ( y ))u x (0, y ) = g1 (y ),
k2 (ϕ1 ( x ))u y ( x,0) = g 2 ( x),
(23)
0 ≤ x ≤ l1 ,
0 ≤ y ≤ l2 ,
0 ≤ y ≤ l2 ,
0 ≤ x ≤ l1 ,
(24)
(25)
(26)
(27)
#∃ ϕ1 (0) = φ1 (0), ϕ1 (l1 ) = φ2 (0), ϕ2 (l1 ) = φ2 (l2 ),φ1 (l2 ) = ϕ2 (0) . &
D = {( x, y ) 0 < x < l1 ,0 < y < l2 }.
1
(23)–(27) (13)–(16). 0 .
). ! − a( x, y )u xx − b( x, y )u yy + u = 0, ( x, y ) ∈ D,
u ( x,0) = ϕ1 ( x), u ( x, l2 ) = ϕ 2 ( x), 0 ≤ x ≤ l1 ,
u (0, y ) = φ1 ( y ), u (l1 , y ) = φ2 ( y ), 0 ≤ y ≤ l2 ,
#∃ ϕ1 (0) = φ1 (0), ϕ1 (l1 ) = φ2 (0), ϕ2 (l1 ) = φ2 (l2 ),φ1 (l2 ) = ϕ2 (0) a ( x, y ) ≥ µ0 , b( x, y ) ≥ µ0 , µ0 > 0,
,
∃
C 2 ( D) ∩ C ( D )
n( x) ≤ ϕ1 ( x) − ϕ 2 ( x) ≤ Ml2 , m( y ) ≤ φ1 ( y ) − φ2 ( y ) ≤ Ml1 , ϕ 2 (x) ≥ 0, φ2 ( y ) ≥ 0, ϕ1 (0) ≥ ϕ1 ( x) + m(0) xl1−1 ,
ϕ2 (0) ≥ ϕ2 ( x) + m(l2 ) xl1−1 , φ1 (0) ≥ φ1 ( y ) + n(0) yl2 −1 , φ2 (0) − φ2 ( y ) ≥ n(l1 ) yl2 −1 , ϕix ( x ) ≥ − M , i = 1, 2,
φiy ( y ) ≥ − M , i = 1, 2, ϕixx ( x) = 0, φiyy ( y ) = 0 ,
− M − φ1 ( y )(2 µ0 ) −1 l1 ≤ u x (0, y ) ≤ −m( y )l1−1 ,
−1
(28)
−1
− M − ϕ1 ( x)(2 µ0 ) l2 ≤ u y ( x,0 ) ≤ −n( x)l2 ,
(29)
m( y ) ∈ C 2 [0, l2 ], n( x) ∈ C 2 [0, l1 ], m "( y ) ≥ 0, n "( x) ≥ 0, M = max{max max ϕix ( x) , max max φiy ( y ) ,
i =1,2
max l2
x
−1
x
i =1,2
y
[ϕ1 ( x) − ϕ 2 ( x)],max l1−1[φ1 ( y ) − φ2 ( y )]} .
x
∀. ! υ ( x,y ) = u(x,y ) + mxl1−1 − φ1 ( y ),V ( x,y ) = −u(x,y ) + φ1 ( y ) − Mx − φ1 ( y )(2µ0 )−1 x(l1 − x ) ,
υ1 (x,y ) = u(x,y ) + nyl2 −1 − ϕ1 ( y ),V1 ( x,y ) = −u(x,y ) + ϕ1 ( x) − My − ϕ1 ( x)(2 µ0 ) −1 y (l2 − y ) .
∋
, υ ( x, y ) − a( x, y )υ xx − b( x, y )υ yy + υ = −φ1 ( y ) + xm( y )l1 −1 ,
υ ( x,0) = ϕ1 ( x) − ϕ1 (0) + xm(0)l1−1 , υ ( x, l2 ) = ϕ2 ( x) − ϕ2 (0) + xm(l2 )l1−1 , υ (0, y ) = 0,
υ (l1 , y ) = −φ1 ( y ) + m( y ) + φ2 ( y ) .
!, # , υ ( x,y ) x = 0 . ,
υ x (0, y ) ≤ 0 , u x (0, y ) ≤ − m( y )l1−1 .
(30)
( , V ( x, y ) , , V ( x, y ) x = 0 . !
Vx (0, y ) ≤ 0 − M − φ1 ( y )(2µ0 ) −1l1 ≤ u x (0, y ) .
% «. &.
∋», ()∗+ 5
(31)
9
3
(30) (31), (28). ( (29). % .
&
4.
!
ϕi ( x) ∈ C 2+α (0, l1 ) ∩ C[0, l1 ],φi ( y ) ∈ C 2+α (0, l2 ) ∩ C[0, l2 ],
i = 1, 2,
h( x, p ) = 0, q (u ) = 1, n( x) ≤ ϕ1 ( x) − ϕ 2 ( x) ≤ Ml2 , m( y ) ≤ φ1 ( y ) − φ2 ( y ) ≤ Ml1 , ϕ 2 (x) ≥ 0, φ2 ( y ) ≥ 0,
ϕ1 (0) ≥ ϕ1 ( x) + m(0) xl1−1 , ϕ2 (0) ≥ ϕ2 ( x) + m(l2 ) xl1−1 , φ1 (0) ≥ φ1 ( y ) + n(0) yl2 −1 , φ2 (0) − φ2 ( y ) ≥
≥ n(l1 ) yl2−1 , ϕix (0) < 0, φiy ( y ) < 0, i = 1, 2, ϕix ( x ) ≥ − M , i = 1, 2, φiy ( y ) ≥ − M , i = 1, 2,ϕ1xx ( x) = 0,
1
1
2
2
−1
, g1 ( y )[m( y )] g1 ( x)[n( x)]−1 – , g '0 , g "0 – . ,
(23)–(27) .
∀. + . .
, − M [1 + φ1 ( y )(2 g '0 ) −1 l1 ] ≤ u x( S +1) (0,y ) ≤ −m( y ) ⋅ l1−1 ,
0 < y < l2 ,
φ1 yy ( y ) = 0, g '0 ≤ − g1 ( y ) − φ1 ( y )l1 , g "0 ≤ − g 2 ( x) − φ1 ( x)l2 , g1 ( y ) < 0, g 2 ( x) < 0, n( x), m( y ) – − M [1 + ϕ1 ( x )(2 g "0 ) −1 l2 ] ≤ u (yS +1) ( x,0) ≤ −n( x) ⋅ l2 −1 ,
0 < x < l1 ,
{
}
) ≤ max {[− g ( x)] ⋅ [n( x)] } ⋅ l .
g '0 M −1 ≤ k1( S +1) (u ( S +1) ) ≤ max [− g1 ( y )] ⋅ [m( y )]−1 ⋅ l1,
y
g "0 M −1 ≤ k2( S +1) (u ( S +1)
−1
x
2
2
, , k1( S ) (u ( S ) ), k2( S ) (u ( S ) ) – , . ,
∃ -
{
}
, u ( S ) ( x, y ) {
}
W p1 2 ( D), ∀p1 > 2 . ! u ( S ) ( x, y ) C1 ( D) . ! (15)
{
}
, k1( S ) (u ( S ) ), k2( S ) (u ( S ) ) -
{
}
C1 [ R1 , R2 ] . #
(13)–(14) u ( S ) ( x, y ) C 2 ( D) . (13)–(15) , s → +∞ , ∃ {k1 (u ), k2 (u ), u ( x, y )} , #∃ (23)–(27)., .
+ (23)–(27) f ( x, y ) #∃ :
l −x
l −y
x
x
y
φ1 ( y ) + 2
[ϕ1 ( x ) − ϕ1 (0) + ϕ1 (0)] + [φ2 ( y ) − φ2 (0) + φ2 (0)] +
f ( x,y ) = 1
l1
l2
l1
l1
l2
y
x
xy
+ [ϕ2 ( x) − ϕ2 (0) + ϕ2 (0)] −
φ2 (l2 ) .
l2
l1
l1l2
)
1. %, 1.1. ∋ /
1.1. %, ./. ∀, 0.!. 4. – 1.: ∋, 1980. – 288 .
2 . 5, 0... / 0... 5. – 1.: ∋, 2009. –
458 .
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17 2011 .
ABOUT EVALUATION OF UNKNOWN COEFFICIENTS
IN A QUASILINEAR ELLIPTIC EQUATION
R.A. Aliyev
1
Inverse problems on restoration of coefficients of the partial differential equation are of interest in
many applied researches. These problems lead to necessity of the approached decision of inverse problems for the equations of mathematical physics, which are incorrect in classical sense. Inverse problems
in definition of unknown coefficients in a quasilinear eliptic equation are studied in the article. Theorems of existence, uniqueness and stability of inversion problems solution for the quailinear elliptic
equations are proved.
Keywords: inverse problem, quasilinear eliptic equation.
References
1. Lavrent'ev M.M., Romanov V.G., Shishatskij S.P. Nekorrektnye zadachi matematicheskoj fiziki i
analiza (Incorrect problems of mathematical physics and analysis). Moscow, Nauka, 1980. 288 p. (in
Russ.).
2. Kabanikhin S.I. Obratnye i nekorrektnye zadachi (Inverse and incorrect problems). Moscow,
Nauka, 2009. 458 p. (in Russ.).
3. Iskenderov A.D. Izv. AN. Az. SSR. Ser. fiz.-tehn. i mat. nauk. 1978. no. 2. pp. 80–85. (in Russ.).
4. Iskenderov A.D. Dif. uravnenija. 1979. Vol. 20, no. 11. pp. 858–867. (in Russ.).
5. Klibanov M.V. Dif. uravnenija. 1984. Vol. 20, no. 6. pp. 1035–1041. (in Russ.).
6. Klibanov M.V. Dif. uravnenija. 1984. Vol. 20, no. 11. pp. 1947–1953. (in Russ.).
7. Khaidarov A. Nekorrektnye zadachi matematicheskoj fiziki i analiza. (Incorrect problems of
mathematical physics and analysis). Novosibirsk, 1984. pp. 245–249. (in Russ.).
8. Vabishchevich P.N. Dif. uravnenija. 1988. Vol. 24, no. 12. pp. 2125–2129.
9. Solov'ev V.V. Zhurnal vych. mat. i mat. fiziki. 2007. Vol. 47, no. 8. pp. 1365–1377. (in Russ.).
10. Vakhitov I.S. Dal'nevostochnyj matem.zhurn. 2010. Vol. 10, no. 2. pp. 93–105. (in Russ.).
11. Lazhyzhenskaya O.G., Ural'tseva N.N. Linejnye i kvazilinenye uravnenija jellipticheskogo tipa
(Linear and quasilinear equation of elliptic type). Moscow, Nauka, 1973. 576 p. (in Russ.).
12. Ivasishen S.D., Jejdel'man S.V. DAN SSSR. 1967. Vol. 172, no. 6. pp. 1262–1265. (in Russ.).
1
Aliyev Ramiz Atash oqlı is Assistant Professor, Department of Computer Science, the Azerbaijan University of Cooperation, Baku.
e-mail: ramizaliyev3@rambler.ru
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