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Journal of Shanghai University (English Edition ), 2005, 9(1) : 20-24 Article ID: 1007-6417(2005)01-0020-05
Nonlinear Riemann Problem for Nonlinear Elliptic Systems in Sobolev Space Wl,p ( D )
SONG J / e ( ~ i $ ) 1 , L I Ming-zhong( ~ ) ~ ~:) 2 1. Mathematics of Department, East China University of Science and Technology, ~umghai 200237, P . R . China 2. Department of Mathema2i~ , College of Sciences, Shanghai University, ~anghai 200444, P. R . China
Abstract The nonlinear Riemann problems were converted into nonlinear singular integral equations and the existence of the solution for the problem was proved by means of contract principle.
Key words nonlinear Riemann problem, nonlinear elliptic systems, singular integral equalions, contract principle. MSC 2000 35J25
1 Introduction
Many problems in mechanics and mechanical
engineering may be formulated into boundary value
problems for first or second order elliptic sys tems,
which were studied by lots of scholars ( see Refs. [ 1 -
9] ) . In this paper we discuss the nonlinear Riemann
problem for nonlinear elliptic systems in the Sobolev
space WI,p ( D ) .
2 Statement of the Problem
Let D ÷ be a domain with a smooth boundary 7, and
F be another smooth boundary containing D ÷ and )'
in its interior in the complex z plane. D - is the
domain closed by F and outside 7. Denote D =
D ÷ + D - "
Problem R Find a solution w ( z ) E W~.p ( D ) , 2 <
p < + 0o to the equation
°w-Foe z ' w ' 7 ~ ' (I)
which satisfies the nonlinear Riemann boundary
condition
w * ( t ) = G ( t ) w - ( t ) + / ~ g ( t , w ) , t E $ , (2)
where w " ( t ) and w - ( t ) a re , as usual , the limiting
values of w ( z ) as z -~ t E y from within D ~ and D -
respectively.
Assumption A
(1) G ( t ) E C ~ ( ~ ) , O < m ~ I G ( t ) [ ~ m 2 , t E T ,
and/~ is certain positive number;
(2) g ( t , w ) satisfies the HSlder-Lipschitz condition
[ g ( t ~ , w l ) - g ( t2,w2)[ ~<H,[ I t l - t 2 [ ' + I w l - w 2 [ ] ,
with the exponent a (1/2 < a < 1) and the constant Hg
related to g ( t , w ) ;
(3) w ( z ) E W x . , ( D ) , F ( z , W , ~ z ) E L , ( D ) , 2 <
p < + oo;
(4) F ( z , w , h ) satisfies almost everywhere in D
the Lipschitz condition
I F ( z , w , h ) - F(z,~,h) I ~<L, I w - w l + L2 1 h - h i ,
while 0 < L2 < 1, and L~ is an arbitrary positive
constant.
3 ]hu~onnation of the Differential Equa- tion into System of Intefrodifferential Equations
Received Sep. 8, 2003; Revised Nov~ 26, 2003 Project supported by the Science Foundation of Shanghai Municipal Commission of Sacience and Technology ( Grant No. 01ZA14023) SONG Jie, Ph .D. Candidate, E-mail: [email protected]; LI Ming-zhong, Prof., E-mail: [email protected]
We define the following singular integral opemtom
of Vekua type E~°~ in D :
1 ~ ' f ( ~ ) d ~ d , ~
I-[~(z) = -~ (~_ z)~orj,
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Vol.9 No.1 Feb.2005 SONG J, et a/. : Nonlinear Riemann Problem for Nonlinear Ellipiic . . . 21
~= ~+i7/ , z = x + i y .
These operators have the following properties in
Lp(D) (see Refs. [ 2 - 3 , 1 0 - 1 2 ] ) :
II T~fII,,D<.Bv t l f l l , , ~ , Bo=2%/--~, l < p < +Qo,
II H~ f l l ~<A, I l f l l , . ~ , A,>~A~=I, l < p < + ®,
3Tar f , 3]-[af 3 f (3) 3Tar I I z f , Tz = 32 3z 3Z - - "
Let w ( z ) E W~,p(D) , I < p < + ao, be an arbi-
trarily chosen general solution of the given partial
differential equation (1) , then we can conclude that it
satisfies the equation
w ( z ) = ~ ( z ) + T~F( ~ , w ( ~ ) , 3 - ~ ) ( z ) , (4)
where q~ E W~.~ (D) and is holomorphic in D.
In the contrary, for the arbitrary holomorphic
function qb ( z ) in D, w ( z ) given in Eq. (4) satisfies
Eq. (1) (see Ref. [4 ] ) . On differentiating the equation (4) partially with
respect to z we obtain the relation
3 w - ~ ' ( z ) + ~[ ,F( ~ , w ( ~ ) 3 - ~ ) ( z ) 3z ' '
and we axe thus led to the following result (see Refs.
I s , m ] ) : Theorem 1 The function w is a general solution of
the partial differential equation (1) if and only if there
exists a function ~ holomorphic in D such that ( w ,
h) satisfies the system of singular integral equations
w ( z ) = ¢ ( z ) + T ~ F ( ' , w , h ) ( z ) ,
h ( z ) = ~ ' ( z ) + I I . F ( ' , w , h ) ( z ) . (5)
4 S o l u t i o n o f I n t e g r a l S y s t e m
In view of the equivalence between (1) and (5) we
just consider the system (5) . Let Op ( D ) denote the
set of all couples ( w , h ) : w , h E L p ( D ) , I < p <
+ ~ . We define the norm in the following way:
II ( w , h) II = II ( w , h) II ,.~ = max(x II w II . . . , II h II . , . ) , X >0.
Thus Op (D) is a Banach space.
For a given in D holomorphic function q~ E
Wl.p(D) , we now use (5) to define an operator P in
O p ( D ) , I < p < + ~ . Suppose ( W , H ) is the image
of ( w , h) E Op (D) under the mapping P
P ( w , h ) = ( W , H ) , W ( z ) = ¢ ( z ) + T ~ F ( ' , w , h ) ( z ) ,
H ( z ) = ~ ' ( z ) + I I D F ( ' , w , h ) ( z ) . (6)
It then follows immediately from (3) that P maps Op ( D ) into itself. Moreover, under certain restric-
lions on the Lipschitz constants L~, L2 and the factor
A, the operator is contractive in Op ( D ). Indeed let
( W, H) and ( W, H ) be the respective images of ( w ,
h ) , ( @, fz ) E Op ( D ) under the mapping P . The
following estimates then hold immediately:
II w - ~¢ I I , ~B~ ( L , II w - ~ I I , + L~ II h - £ II ~)
= B~L, ( ~ II w - ~ II , ) + B ~ L ~ II h - £ II
~<B~(L~ + ~ L ~ ) m a x ( ~ II w - ~ II , , II h - £ II , )
=BD(L, +AL2)II ( w - W , h - £) II ,.~ <~Ba(L~ + AL2) II ( w , h ) - ( ~ , £ ) II p.~, (7)
II g -/-/II p
~<Ap(L, II w - @ l ip+L2 H h - / ~ l l p )
~<Ap( ~-L,1 + L2)Hh21X( A II W -~) II p, II h - f~ II p )
=A,(-~L. + L,) ll ( w - ~ , h - £ ) ll,.~
(1 ) (8) ~<A, ~-L,+L~ l l (w,h)-(~,,h) l l , . , ,
According to (7) and (8) , we have
II ( W , H ) - ( V¢,I-I) I1,.~
II ( w , h ) - (Cv , £ ) II ,,,
< max(Ap ,ABD) ( 1 ~-L, + L 2 ) "
II ( w , h ) - ( ~ , £ ) II ,.~.
Denote 0 = m a x ( A , , A B D ) ( 1-~L, + L2) , (9)
then
II ( W , H ) - ( ~ f , i t ) I I ,.~ < o II ( w , h ) - (~v , £ ) II ,.~. (10)
If we choose L~ and L2 sufficiently small, we have
0 < 1. It thus follows from Banach's fix point theorem
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22
that P has a unique fixed point ( w , h ) ~ Op ( D ) ,
namely,
T h e o r e m 2 Under Assumption A, if the inequality
(10) holds, the operator equation (6) has the unique
fixed point ( w , h ) for arbitrary given holomorphic
function q b ( z ) ~ W~,~(D), and w ( z ) 6 W~,~(D) is
the unique solution to Eq. (4) or Eq. (5) .
5 T h e N o n l i n e a r R i e m a n n B o u n d a r y V a -
lu e P r o b l e m
It has been known that any general solution w of
the partial differential equation ( 1 ) takes the form
( 4 ) , so we shall exploit the arbitrariness of qo by
choosing it in such a way that w then satisfies the
jump condition (2) .
On writing q) as a sum of two holomorphic functions
qb and q5(~,~) it follows from (2) and (4) that
=/~g + G[ ToE] - - [ TOE]*, t ~ ~'. (11)
Obviously as F C L~ ( D ) , p > 2, ToF is a HOlder-
continuous function in the whole plane (see Refs. [ 2,
10]) , then
[ T~FJ ÷ = [ T,F] - , t 6 9",
and the Riemann boundary value problem (2) for w
can therefore be resolved into two such problems for
the holomorphic functions ~ and ~(w.~)
(P; ( t ) - G ~ ; ( t ) = ~ g ( t , w ) , (12)
~(~.h> ( t) -- CrcP(,~.~)( t) = g(,.~)( t) , (13)
where
a(~.~>(t) = ( G ( t ) - 1) T o E ( ' , w , h ) ( t ) ,
• [ , ( t ) = ~ ; ( t , w ) , c p ; ( t ) = ¢ ; ( t , w ) .
The nonlinear Riemann problem (12) has the solution
(see Ref. [ 14 ] )
Journa of ~ . n a z i Vn/~s/ty
~ o ( z ) = ~ o ( z , w ) _ X ° ( z ) I l~g( t 'w) dt 2~ri X" ( t ) ( t - z ) +
7 X , ( z ) P . ( z ) , (14)
where X u ( z ) is the canonical solution of the Riemarm
problem, n = index( G, )') I> 0,0 < mg ~< X u ( z ) ~< M u ,
and P. ( z ) is an arbitrary polynomial of degree n .
If n = index( G, )') <~ - 1,
~ , ( z ) = ~ , ( z , w ) =
- . - I dt Xo(z ) ( p g ( t , w ) + ~ ; t ~ t ' ) x . ( t ) ( t _ z ) ,
n< - 1 , X~(z) l~g ( t ,w )d t --~-~-- I X+ ( t)( t _ z) " 7
n= - 1 .
where the parameters 2j ( j = 1 , 2 , - " , - n - 1) are the
solutions to the following algebraic system:
- . - I f k - j - I aj ~ d t = t ~ g ( t ' w ( t ) ) f - t d t , j r , X ( t ) - X + ( t )
9" k = 1 , 2 , . . . , - n - l ,
and w is the generalized solution of Eq. (1) . Define an operator L in a manner similar to the
mapping P defined earlier. For a given couple ( w , h)
EOp(D) ,2< p< oo , w e s e t
L ( w , h ) = ( W , H ) , W(z ) = qbg(z) + qb(~,~)(z) + T o F ( ' , w , h ) ( z ) , H ( z ) = qb;(z) + ~(,,~) ( z ) + ] - [ o F ( ' , w , h ) ( z ) ,
(15)
where ~Pg and qb(~.h) are the holomorphic functions
defined and constructed in the preceding section. It is
obvious that L maps Op (D) into i t se l f .
Suppose ( W, H ) , ( I~,/-/) be the respective images
of ( w , h ) , ( ~b,/~) E O~ ( D ) under the mapping L ,
similarly to the above (see Ref. [ 10 ] , Chapter One ),
we can obtain the estimate
II w - fe II p <<.a II v , ( z , w ) -
)tB. ( L~ II w - ~ II p + Le 11 h - h II ~ )
~ I ~ M g H II w - ~ v II +~ M(w'h)MBo(L, II w - ~ b II mg a P m(w.~) P
2So ( L~ it w - ~ it p ÷ L2 II h - £ It ~)
+L2 t[ h - h [ J p ) +
M(w,a) ) I ~MaH + MBDL I+BoL ~ (3, l[
m a a m(w,~) P '/T~(~,h ) P
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Vol.9 No.1 Feb.2005 SONG J, et a/. : Nonlinear Riemann Problem for Nonlinear Elliptic . . . 23
[ ( M(~, ~,) ) <~ l zMsH +M(~'h) MBoL~+BDLI+IBoL2 ' M + I
ms s .D%(w,h ) m(w.h)
<<. t zMsH +M('~'~)MB~L~+B~L~+ABDL2 ' M + I
..~ms( M~ , + MsCHs) ii w _ ~?v ll ~, + !l~'~a,) MBD( L~ t] w _ ~ ]] p + L~ ll h _ f ~ ll ~)
M('~'~')MCBv(L~ [I w - ~ I[ + L 2 [I h - f ~ [ l ~ ) + A ~ ( L 1 [[ w - ~ I[ +L~ [[ h - £ 1 [ ~ ) m(w.~) ~
<~ ( M, Hs + MsCHs) + !t~'~,~) MBoLt + " MCBoLt + A,L~ + m(w,~) m(~.~)
~'~,~> MBoL2 + M(,~.~,) M'A~L~ + ApL2] max( A [[ w - ~ [] p, [[ h - £ H )
<~ ( M~,Hs + MsCHs) + M(',~,~ MBoL ~ + MCBoL, + A,LI +
~'w'~)MBDL2 + M(~'~) M'A~L~ + A~L2] H ( w , h ) - ( ~v fz) l] ~,~, m( w,~) ~tg( w,~)
MB~Lz + M(w.,) M'A~,L~ + A~L2] ]] ( w , h) - ( ~v, h ) I] ~.~, m(w,~) m(w,~)
where I G ( t ) - l l <. M. Denote
m a x ( ,~ II w - ~ II , , II h - £ II , )
] l l . ( w , h ) - ( ~ , h ) I1,.,,
, + A , ( L , II w - ~ II ,. + L2 II h - h II ~, )
M( ,~,h) MCBoL1 + ApL1 }
Tfg( w.h) +
( ) f l = t ~ M s H +M(~'h) MBoLI+B~LI+ABoL2 Mc~'h) M+ 1 +-~ II~'Hs+MsCHs)+ ms a m(,~.h) m(,~.h)
MBDL ~ + M(,~.t,) MCBDL~ + ApL~) + ~',~,h) MBDL 2 + M(,~.h) M'ApL2 + ApL2,
SO
II ( W , H ) - ( V ¢ , [ - I ) II , . , ~ , e II ( w , h ) - ( ~ , £ ) II , .~.
If /~ and L~, L 2 are sufficiently small , and
0 < fl < 1, (18)
thus we have the following conclus ion by means o f the
cont rac t principle :
Theorem 3 Under Assumpt ion A, w h e n n =
index( G , ) ' ) ~>0, if the inequality ( 1 8 ) holds , the
problem (1) and (2 ) have the solution w ( z ) E Wl.p
( D ) (2 < p < + ~ ) , which depends on n + 1 arbitrary
real constants .
In a similar way , we can also obtain the following
theorem :
Theorem 4 Under Assumpt ion A, when n =
index( G , ) ' ) ~< - 1, if the inequality (18 ) holds , the
(16)
(17)
problem ( 1 ) and ( 2 ) have the solution w ( z ) E
W~.p(D) ( 2 < p < 0o ) under the sufficient and
necessary condit ions tha t the parameters ~ sat~fy the
equalities
- n-I tk-J -I "= ( t ) '
k = l , 2 , . . . , - n - 1 .
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