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: songjie@ecust.edu.cn; LI Ming-zhong, Prof., E-mail: mzhii@publicS.sta.net.cn

We define the following singular integral opemtom

of Vekua type E~°~ in D :

1 ~ ' f ( ~ ) d ~ d , ~

I-[~(z) = -~ (~_ z)~orj,

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

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

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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