Research Journal of Applied Sciences, Engineering and Technology 4(8): 884-887, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: October 07, 2011
Accepted: November 02, 2011
Published: April 15, 2012
On Uniqueness of Strong Solution of Dirichlet Problem for Second Order
Quasilinear Elliptic Equations with Corsed Condition
Seyed Ahmad Hassanpour
Islamic Azad University, Babol Branch, Babol, Iran
Abstract: The first boundary value problem is considered for second order quasilinear elliptic equation of nondivergent from when the leading part satisfies the Cordes condition. The uniqueness of strong (almost
everywhere) solution of mentioned problem is proved for any n$2.
Key words: Boundary value, quasilinear, strong solution, uniqueness
INTRODUCTION
equivalence and nonsingular linear transformation in the
following sense: domain D can be covered by a finite
number of subdomains D1, …, Dl so that in each Di, i =
1, …, l, Eq. (1) can be replaced by the equivalent equation
L' u = f '(x), and nonsingular linear transformation of
coordinates can be made, at which the coefficients of the
image of the operator L' satisfy condition (4) in the
domain Di.
The aim of the present study is to prove the
uniqueness of strong (almost everywhere) solution of the
first boundary value problem (1)-(2) for f(x) , Ls (D) at
s , (n, 4) for any n $2.
Note that the analogous problem was considered in
(Chicco, 1971) for n = 2,3,4 and the strong (almost
everywhere) solvability was proved for f(x) , Ls (D) at
s ,[2, 4), for n = 2,3; s,[1, 4), at n = 4. in this connection
we refer to the papers (Ladyzhenskaya and Ural'tseva,
1969; Chicco, 1971), where the analogous results for the
second order linear elliptic equations with continuous
coefficients were obtained, and we also refer to the papers
(Talenti, 1965; Mamedov and Agayeva, 2002; Vitanza,
1992; Vitanza, 1993; Alkhutov and Mamedov, 1985),in
which some classes of above mentioned equations with
discontinuous coefficients were considered. We notice the
papers (Wen, 2000), where the questions of strong
solvability of boundary value problem for the second
order parabolic equations were investigated. Existence of
strong solution of the first boundary value problem (1)-(2)
was established in (Wen, 2000), at that it was done for
more general class of equation than (1).
We now agree upon some denotation. For P,[1,4) we
denote by Wp1(D) and Wp2(D) Banach spaces of function
u (x) given on D with the finite norms:
The aim of this study is to prove the uniqueness of
strong (almost everywhere) solution of the first boundaryvalue problem for second order quasilinear elliptic
equation.
Let En be n-dimensional Euclidean space of points
x= (x1, …, xn), n $2 , D be a bounded convex domain in
En with the boundary MD,C2.
Consider the following Dirichlet problem in D:
n
Lu 
 aij ( x, y)uij  f ( x), x  D;
(1)
i , j 1
u| D  0
(2)
Where
ui ui 
u
 2u
, uij 
; i , j  1,..., n;
xi
xi x j
ai j ( x , z )
Is a real symmetric matrix whose elements are measurable
in D for any fixed z 0E1 , moreover:
 | |2 
n
a
i , j 1
ij
( x , z)i  j   1 ||2 ;
(3)
x  D, z  E 1 ,   E n
n
 aij ( x, z)
  sup x D, zE1
i , j 1
 n

 aij ( x , z ) 
 i 1


2

1
.
n 1
(4)
u
Here :0(0,1] is a constant. Condition (4) is called the
Cordes condition, and it is understood to within
1
( D)
Wp




n

 

 | u| p 
|ui | p  dx

D

 
i , j 1

and respectively,
884

1
p
Res. J. Appl. Sci. Eng. Technol., 4(8): 884-887, 2012
u
Wp2 ( D)




n
n
 


 | u| p  | u | p 
|uij | p  dx
i


D
 

i 1
i , j 1



Theorem 2: (Ladyzhenskaya and Ural'tseva, 1969).
Let the coefficients of the operator M satisfy conditions
(3'), (4'), (5').
Then there exists p1(:,F , n) 0(,2) such that any p ,
[p1,2] for any function u(x) , êp2(D) the following
estimate holds:
1
p
Further, let Wp01(D) be a completion of C04 (D) by the
norm of space Wp1(D), and êp2(D) = W22(D)1êp1(D).
Function u(x) 0 êp2(D) is called strong solution of the
first boundary- value problem (1)-(2)(at f(x) , Lp(D)) if it
satisfies Eq. (1) almost everywhere in D.
Everywhere below notation C(…) means that positive
constant C depends only on what in parentheses.
We now give some known facts which we will need
further on:
| aij ( x , z 1 )  aij ( x , z 2 )|  H (| z 1  z 2 |;
x  D; z 1 , z 2  E1 ; I , J  1, ..., n
If p > n, then:
Here ux = (u1,…,un):
Consider the following equation in D:
n
 aij ( x, u, ux )uij  f ( x), x  D
Theorem 4: (Chicco, 1971)
Let n = 2, and let the coefficients of the operator M satisfy
conditions (3') and (5) (for " = 1 then for any s 0(2, 4) at
A>0 there exists DA = DA (:, MD, H , s, A) such that if
mesD#DA, f(x) 0 Ls(D) and || f || Ls ( D) # A then the first
boundary value problem (1)-(2) has a unique strong
solution u(x) 0 ê22(D).
(1')
i , j 1
and suppose that the elements of the real symmetric
matrix ||aij(x, z, L)|| are measurable in D for any fixed
z,E1, L 0 En:
 || 
 aij ( x, z, )i j  
(7)
with some constant H1$0.
Then for any s 0(2, 4) at n = 4; for any s 0[2;4) at n
= 3 and A>0 there exists *A = DA (:, F, 0, MD, H1, s, A)
such that if mesD#DA, f(x) 0 Ls(D) and || f || Ls ( D) #A
then the first boundary value problem (1)-(2) has a unique
strong solution u(x)0W2 (D).
u L ( D)  C2 ( p, n) u x L p ( D)
n
(6)
Theorem 3: (Chicco, 1971)
Let 3 # n #4, and the coefficients of the operator L satisfy
conditions (3)-(4) and:
u L p ( D)  C1 ( p, q , n) u x L p ( D)
2
W p
at that for any A>0 there exists dA = dA(:,F, n, MD, H, ",
A) such that if mesD,dA, then the first boundary value
problem (1')-(2') has a strong solution from the space
ê22(D) for any function f(x) , L2(D), whenever || f || Ls
( D) #A.
Theorem 1: (Ladyzhenskaya and Ural'tseva, 1969)
Let 1<p<n, 1# q # np/n-p. Then for any function u(x) ,
êp1(D) , the following estimate holds:
Mu 
2 ( D)  C3 (  ,  , n,  D) Mu L p ( D)
u
1
|| ;
Theorem 5: Let n$5, and the coefficients of the operator
L satisfy conditions (3)-(4) and (7). Then for any s 0(n, 4)
at A>0 there exists DA = *A (:, F, 0, MD, H1, s, A) such that
if mesD#DA , f(x)0 Ls(D) and || f || Ls (D) , A then the
first boundary value problem (1)-(2) has a unique strong
solution u(x) 0 ê22(D).
2
i , j 1
(3')
x  D, z  E1 ,,  En ,   En ,
n

a
x  D, sup , u  E n
z E1
i , j 1
ij
( x , z, v )
 n

  aij ( x , z, v ) 
 i 1

2

1
.
n 1
Proof: The constant D A will be chosen such that pA DA
#dA. Therefore, according to theorem 2 we must prove
only the uniqueness of the solution. Let u1(x) and u2(x) be
two strong solutions of the first boundary problem (1)-(2)
from the space ê22(D), and we have:
(4')
and beside this
n
|aij ( x , z1 ,1 )  aij ( x , z 2 , 2 )| H (| z1  z 2 |a  |1   2 |a )
x  D; z1 , z 2  E1; 1 , 2  En ; i , j  1, ..., n
L(1) 
(5)
a
i , j 1
ij
( x )u (1) ( x )
2
 xi  x j
n
n
i , j 1
i , j 1
L(1) (u 1  u 2 )  aij ( x , u 1 )uij1   [aij ( x , u 1 )
with constants H$0 and a , (0,1].
 a ij ( x , u 2 )]uij2  f ( x )
885
Res. J. Appl. Sci. Eng. Technol., 4(8): 884-887, 2012
Proof: Let n$3, and u1(x) and u2(x) be two strong
solutions of the first boundary value problem (1)-(2) from
the space ê22(D) and:
n


[aij ( x , u1 )  aij ( x, u 2 )] uij2  F ( x )
(8)
i , j 1
On the other hand, according to (7):
n
M (1) 
i , j 1
n
| F ( x )|  H1 | u 1  u 2 |  | u 2 i j |
(9)
We have:
i , j 1
Let q1 = np1/n-p1. From theorem 1 we obtained:
u 1  u 2 Lq 2 ( D)  C1 ( p1 ) u 1  u 2
( D)
1
W q1
n
a
M (1) (u 1  u 2 ) 
(10)
i , j 1
( x , u 1u 2 , u 1x )uij1
ij
n

 [a
i , j 1
Now applying theorem 2, from (8)-(10) we conclude:

ij
( x , u 1 , u 1x )  aij ( x , u 2 , u x2 )] uij2  f ( x ) 
ij
( x , u 1 , u 1x )  aij ( x , u 2 , u x2 )] uij2  F ( x )
n
 [a
i , j 1
u 1  u 2 Lq1 ( D)  C1C3 F L p1 ( D) 

 n

 C1C3 H   | u 1  u 2 | p1   | uij2 |
 i , j 1 
 D
2
 aij ( x, u1( x), u1x ( x)) xix j
(12)
1
p1
 p1
dx  

‘On the other hand, according to (5):

F1  x   H u 1  u 2  u 1x  u x2
1
n
  n
n

 C1C3 u 1  u 2 Lq1 D      | uij2 | dx  
 D   i , j 1 
 

Let q2 
n
 u
2
i, j
i, j  1
(13)
.
np1
n  p1
From theorem we obtain:
 2nC1C3 H u 1  u 2 Lq1 ( D) u 2 W22 ( D)(mesD)
 2nC1C32 H u 1  u 2 Lq1 ( D) f L2 ( D)(mesD)
 2nC1C32 H u 1  u 2 Lq1 ( D) f Ls ( D)(mesD)
 2nC1C32 HA(mesD)
 2nC1C32 HA A
s n
ns
s n
ns
2 n
2n
2 n
2n

u1  u 2
 C1  p1  u1  u 2

Applying theorem 2, from (12)-(14) we conclude:
(11)
u1  u2
u 1  u 2 L q 1 ( D) 
Wq12
( D )  C1C3 F1
L p1 ( D )
u 1  u 2 L q 1 ( D)
1
p1

 p1

n
p1



2

 C1C3 H  | u1  u2 |  | u1x  ux2 |
| uij
| dx 


D

 i, j 1 



s n
2s

1
2

2
C1C3 H u  u |W 1
q 2( D )
n

n

2
| uij
| 


 D i, j 1 

2 n
 2 nC1C3 H u1  u2
Wq12 ( D )
u2
Wq22 ( D )
( mesD ) 2 n 
s n
 2 nC1C32 H u1  u2
u1  u 2 Lq1 ( D) 

1
1
We choose D = min{dA, D'}. Then from (11) it follows
that:
1 1
u  u 2 Lq1 ( D)
2
Wq12 ( D )
u2
( mesD ) 2 s 
Ls ( D )
s n
 2 nC1C32 HA Asn
u1  u2
i. e., u ( x )  u ( x )a. e.inD.
1
(14)
Wp21 ( D )

2  n s 2
2n 2s
Let D ' be such that:
2C1C32 HA(   )
Wq14 ( D)
2
Theorem 6: Let n$3, and the coefficients of the operator
M satisfy conditions (3')-(4') and (5) at "=1 then for any
s0(n,4) at A>0 there exists D A = D A (:, M D, H , s, A) such
that if mesD,DA, f(x),Ls(D) and || f || Ls (D) #A then the
first boundary value problem (1' )-(2) has a unique strong
solution u(x), ê22(D).
Wq12 ( D )
Let D'' be so that:
 2nC1C32 HA(   )
886
sn
sn

1
2
(15)
Res. J. Appl. Sci. Eng. Technol., 4(8): 884-887, 2012
We choose D = min{dA , D’}. Then from (15) it follows
that:
u1  u 2
1
W q 2 ( D)
i.e.,

1 1
u  u2
2
Mamedov, I.T. and R.A. Agayeva, 2002. The first
boundary value problem for non divergent linear
second order elliptic equations of cordes type. Trans.
Nat. Acad. Sci. Azerb,22 (1): 150-167.
Talenti, G., 1965. Sopra una class di equazioni ellitiche a
coefficienti misurableili. Ann. Mat. Pura Appl., 69:
285-304.
Vitanza, C., 1992, W2,p regularing for a class of elliptic
second order equations with discontinuous
coefficients. Le Matemetiche, 47: 177-186.
Vitanza, C., 1993. A new contribution to the W2, pregularity for a class of elliptic second order
equations with discontinuous coefficients. Le
Matematiche, 48: 287-296.
Wen, G.C., 2000. Initial-mixed boundary value problems
for parabolic of second order with measurable
coefficients in a higher dimensional domain. Proc. of
the Second ISAAK Congress, 1: 185-192.
1
W q 2 ( D)
u 1 ( x )  x 2 ( x )a. e.inD
The theorem is proved.
REFERENCES
Alkhutov, Y.A.and I.T. Mamedov, 1985. Some properties
of the solutions of the first boundary value problem
for parabolic equations with discontinuous
coefficients. , Sov. Math. Dok1., 32(2): 343-347.
Chicco, M., 1971. Solvability of the Dirichlet problem in
H2,p ( ) for a class of linear second order elliptic
differential equations, Boll. Un. Mat. Ital., 4(4):
374-387.
Ladyzhenskaya, O.A. and N.N. Ural'tseva, 1969. linear
and Quasilinear Elliptic Equations. Academic Press,
New York.
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