Applied Mathematics and Mechanics (English Edition, Vol. 13, No. 1, Jan. 1992)

Published by SUT, Shanghai, China

SINGULARLY PERTURBED SEMILINEAR ELLIPTIC EQUATION WITH B O U N D A R Y - I N T E R I O R L A Y E R I N T E R A C T I O N Zhou Zhe-yan ( ~ ) (Department of Mechanics, Fujian Normal University, Fuzhou )

(Received Nov. 13, 1989; Communicated by Lin Zong-chi) Abstract In this paper, we consider the phenomenon of the boundary and interior layer interactionsfor a class of semilinear ellipticequation. Undersome appropriate conditions. weget the existence of the exact solutionfor the problem and its high order uniformly valid expansion. Key Words boundary layer, interior layer, semilinear elliptic equation

I.

I n t r o d u c t i o n a n d t h e C o n s t r u c t i o n o f Our E x p a n s i o n s

The phenomenon of interior nonuniformities in solution or its derivatives of singular perturbation problems has been extensively studied by several authors (see, for example, Jiang Furu t~land K.W. Chang and F.A. Howest2-51and many references contained therein). Those results are mainly within the case of ordinary differential equations. The problem involving turning point for the second order ordinary differential equations was completely investigated by [1]. K.W. Chang and F.A. Howes, however, used differential inequality theory to study those singularly perturbed problems for some nonlinear types. For the case of partial differential equations, some analogous results have also been achieved. Works by E.M. de Jager, for example, dealt with the intersection layer problem for a class of linear elliptic equationt6J. See also F.A. Howes [71,whose works are of importance since it concern nonlinear equations. But they did not give the high order asymptotic expansion for the problem. In fact, compared with the case of ordinary equations, works on this subject are rarely seen, so we may say that such a study is just at beginning. We are interested in a problem of the form eAU=h(X,U) (XE~2) (1.1)

f

l U:q~(X)

(XEO0)

(1.9.)

where A is Laplace operator, O is the domain with x ~+ yZ< 1 ( X : ( x , y ) ) and e is a positive parameter. We also use O+ and O_ to denote the subdomain with y>O and y < 0 in ~Q respectively so t h a t F : { ( x , O ) j - - l < x K l } is a boundary line of O+ and ~Q_ in ~Q. Also, qJ(X)ECWt.*'(O~) and D : ~ ( x , y , U ) j UER, (x,y)E~} Let h(X U) in (1.1) satisfy the following conditions: i) h(x, y, U) is sufficiently differentiable with respect to x and u in D, and to y in~2§ • Rand 77

78

Zhou Zhe-yan

_ x R, respectively, (1.3) ii) h~fx, y, U ) = I > 0 , for (x, y, u) t5 D. (1.4) When e-----0 , equation (1.1) reduces to h( X , U ) = 0 . Our work in this paper mainly is based on the supposition that the equation h(X, U)-----0 has a solution U0(X) which-satisfies i) Uo(X) is continuous in ~ , ii) Uo(X) is sufficiently smooth with respect to x in ~ and with respect to y in ~+ and ~ _ , respectively. THis means that OU,/Oy may intersects a l o n g / " in g2. First, we construct the out expansion for (1.1)-(1.2) and denote it as the form oo

U(x,y,e)~.~,,(x,y)e"

for((x,y)15~§

(1.5)

for ( ( x , y ) 1 5 ~ _ )

(1.6)

oo

U(x,y,e).-~"]~.(x,y)e" fl-0

It can easily checked that the terms

h(x,y,eo(X,y)

tZ, satisfy

)=o

t,.(x ,y ,~,(x ,y) ) .~.(x, u ) = t~u._ ~(x ,y) + g.(x,y)

(1.7) (1.8)

and

h(x,u,n0(x,y))=0 h.(x,y,n,(x,u))n.(x.u)--txn,_,(x,u) + #.(x,u)

(1.9) (1.10)

Where the function ~. (x ,y) ( iff. (x ,y) ) is,, determinable from the previously found function. ~o ,~1 ,"" , ~ . - l ( g o ,at , " " ,~.-1 ) Let ~o ( x , y ) = u o ( x , y ) for (x ,y )15g2+ ) and ~ 0(x ,y ) = u0 (x ,y ) for (x ,y )E~2., respectively. B3~virtue of(l.4), all the terms ~. and ~l,(n~.~l) can uniquely be determined. Thus, we get the out expansion as the form

_ . 1 7 ( x ,y ,~

U~ II.

( ( x ,y )15f~ +)

((x,y)6~_)

(1.11)

The Constructi o f the Interior E x p a n s i o n s

In this section, we shall introduce a kind of inner expansion to destribe the interior layer phenomena for the problem. Let t----y/M'-~ and the inner expansion be the form oo

V(x,t, ,,/-7) N ~ ~.(x. t)~k

(t>~o)

n--0

Y'(x , t , , , / - ~ ) =

(2.1)

P'(x,t ,,,~)-..~ ~.(x, t)e}

(t~o)

IQ--O

In the new coordinate system (x, ~), Laplace operator eA can be rewritten as follows 0~ 02

eA = - ~ r + e ~ - r

(~.2)

Singularly Perturbed Semilinear Elliptic Equation

79

Substituting U(x ,y ,e)+17 (x ,t, ~/"i-) and ~ ( x ,y ,e)+ ~ ( x ,t, ~/e'-) in to equation (1.1), we get from (2.2) that for t ~ O _. Ozg Otg(x,f , # - T ) _ _ h ( x ,t~/%-',U + V) - h ( x , tM-'~e , u)--e-O-~ Otz

(2.3)

and

O ~ ( x ' t ' -M-~')----h(x,t~/-~,U + ~ ) - - h ( x , t ~ ' e - , Otz

In order that Uout(x ,y ,e ) + V (x ,t, ~

~ ) - - e ~ x 2 for t-.<0

(2.4)

) can be continuously differentiable on./-', it requires

that

jU(x ,v :)1 ,.o + r otT F

[o. t,-o

+

1

,t, # - 7 ) i,.o--O (x,v : ) i ,.0 + P ( x , l ,~) 1,.o og - -

I

o,

t,-~

=

oU ~ - o. I,-o

+

~

1

.,

-

-

o~

(2.~)

(2.6)

I,.o

By comparing the same order of the power of ~-~'e, we get 9

~o(x,O)+Oo(X,O)=~o(X,O)+~o(x,O)

[ o,(x,o)--~,(x,o) [ ~,(x,o)+odx,o)=~,(x,o)+~dx,o)

]

o~(~,o)=~,(~,o)...

(2.7)

/ ~.(x ,o ) + o,.(x ,o ) =n.(x ,o) + G.(x .o ) /~

~.+,(x , o ) = -~ ~ . §

OOo(X,O) Ot oa~ Oy

( n= , ,0 1 2 , ..-)

xO

Oeo(X,O) Ot _~

o o , ( x , o ) = Ono(X,o) ~ O~dx,O) Ot

8y

Ot

o~dx,o) _ o e , ( x , o ) Ot Ot a~,(x,o) Oy

~ Oo&x,O) = o ~ . ( x , o ) Ot 8y

(2.8)

~ O~dx,o) 8t

o o . . ( x , o ) _ o G . ( x , o) Ot Ot •

+

Oy

oo,,+,(x,o) =. o~,(x,o) + OG,+t(x,O) 8l

8y

From (2.3), (2.4), we see that oo , equation .fafnilies

l --girOZOo = h ( x , O, ~o(X, O) +

Or,

Ot

(n=0,1,2,-")

02 ( ~ 0 , ~ t , ~ z ) satisfy the following differential

- h(x ,0 ,~*(x ,0 ) )

(t>~o)

0~ o = -yF-=h( x,O ,uo( x,O ) + o ,) - h ( x,O ,no( x,O ) )

(t~o)

Oo)

-

(2.9)

80

Zhou Zhe-yan

"''X'-

{ 0201

'

" - ~ W = _~ ( )v~

(t>~o)

~

(t~0)

---K(xI:J,

O~Vt

(2.10)

(t>0)

- T i r : K ( x )~z + O,( x ,t )

(2.11) -~'

=K(x)~,

+ G,(x ,t )

(t~0)

Generally, for n/>3 , there are

I--ow=K(x)e.+O.(x,O O2~

(t>~O) (2.12)

O n / " , ~0(x,0)----~0(x,0), and by (2.7), (2.8), we may set vo(x,t)----~o(x,t)--O K ( x ) = h , ( x ,0 ,~o(x,O ) ) = h~(x ,0 ,~o(x ,0 ) ). Then formulars (2.10) - (2.12) follow.

9 Let

The solutions to equation (2.10) with boundary conditions (2.7), (2.8) are easily determined as follows:

I

.

v.... ,(x,

.,

1

[ 0~0(x,0)

r~ = - Z E ~ -~( x . ~ ,

i o~o(~,o)~ au " _ a~o~y,o).)~xpE./-X-~T(.~) ,2

(,
(2.13) Because K(x)>~l:>o , both e l ( x , t ) ( t ~ 0 ) and ~ l ( x , t ) ( t ~ O ) are of boundary layer type, namely, o , ( x , t ) goes to zero exponentially as t ~ + c o , uniformly in x and so. does. ~ l ( x , t ) as t ~ - - o o . The general solution of (2.11) can be expressed by ] i

o,(x,t)=ez(x)oxp[-,,/K----(~-t]-2,v/~[Io

exp[M(&~x)( r - t ) ]

9

~,(x,t):~,(x)exp[~/K(x) t]=

9G~(x, , ' ) d r -]-

I'

1

2M~[I,

".

O

exp[~,/~(t--r)]

=

expC,,/K-R--(-~-(r--t)]adx, ,:)dr

]

(2.15)

-oo

It's easy to verify that . ( ~ z ( x , t ) = O ( $ z ( t ) e x p [ - ~ / K ( x ) t - I ) ( as t-->+oo ,uniformly inx)and G~(x, t ) - - - - - O ( ~ z ( t ) e x p [ ~ t ] ) (as t-->--oo , uniformly in x), where ]~z(t) and ~ ( t ) are both polynomials of t. F r o m (2.13), (2.14), and the previous discussions, ~z and Y2 are both no doubt of

Singularly Per~iarbed Semilinear.Elliptic Equation

81

boundary layer type. To determine ~ ( x ) and ~2(x), we may refer to (2.7) and (2.8) and get 1

1 0 2,,/'h"(~-J.~ expl:~/~r]

1

+I [+'~ 2~/(~/~-~) j~ e x p [ _ - ~ / K ( x )

e'(x)----T[n'(x'~176 9.G,(x,r)dr -

(2.16)

+,.(x)=-~EO,(x,O)-n,(x,O)]9 O~(x,r)dr

r3 (2". 17)

By the method of {nduction, we are able to prove that G,(x,~)=O(),(t)expE, , / ( ~ - ~ t ] ) ( ast ---> + ~ ) and G.(x,t)=O(~(t)exp[~/'-K--~ t3)( as t - - - > - ~ ) . T h e functions ~ , ( t ) and ~ , ( t ) are polynomials with their degrees less than n. This enables us to solve all the equations in (2.12) associated with their boundary conditions (2.7), (2.8), and those solutions possess the form similar to (2.14) and (2.15). III.

The C o n s t r u c t i o n

Boundary Layer

of the

Expansions

The boundary layer phenomena may occur because our out expansion and inner expansion are independent of the boundary conditions of the problem. To describe this, we need to construct boundary layer function near the boundary of the domain. Let

0~Q1----0/'2f] Q§

. Then operator eA becomes

and .~----~ - - - - - Y ~A=L0

+ ~/'T L~+eL,

(3.1)

where

I

L~

i

03

-Y~"

2x L,= -,~l--l_x,

Oz OxO~

(3.2)

L~--~-~.. Ox along 0~1 is in the form

Suppose the boundary layer function Z ( x , r co

n

(3.3)

~( x ,~ , # - g - ) N Y2, z.( x ,~ )e ' IImO

and substitute ./7 + Z to (1.1), Then we get 1

OqZ~O

1 --X 2

O+ 2

..

(3.4)

1-----~x - b ~ =ntx' ~/T-s~,~o(X, l~/Y-z~--x~)+~0)

-[h,(x,~/l-x -- [-h,Cx,,v~,

and generally for n ~ 2

we get

~ ,~o+~o)-h,(x, ~0+~0)-h,(x,

~/-i-x ~ , ~0)3 ~--~~.~ ,v / 1 - x

2 ,u0)-]'+-L,z0

(3.5)

82

Zhou Zhe-yan

1-x' ~n.~x,~q--~-,Uo(X,

J ~

+ ~o)~. + lr.(x,~)

(3.s)

where llr,,(ac,~)(n-----0,1,2, -.-) is only dependent on ~ and Uwith i~n--1 An essential r~quirement is that /7 + Z satisfy the boundary condition of the problem on 0Or, namely,

E/7(x,u,e) + Z(x,L d%")7o~, ~ ( x ,

~/T's~) =~(x)

(s.7)

From (3.7) we find

~(x,0)f~(x)-~0(x,,

[

q--'A~)

~t(x,0)=0

""

(s.8)

~,,_t(x,0)=0

~.(x,0)ffi-~.(x,

~/is~-r)

According to (3.4) and (3.8), we have i) if ( z ( x ) - - - - - 4 P ( x ) - - ~ o ( x , ~ / ~ ) ~ 0 ,

(mffil,2,.-.)

then

r*c=)

dr/

ii) if a ( x ) - - - - - ~ ( x ) - - ~ o ( ~ , ~ - ~ ' Z ' ~ ) < 0 , ~ = i~o

_

(3.9)

then dr/

a(~

)+r)dr

(3.10)

iii) if a(x)ffi0, then z , ~ 0 . L e m m a 1 Let a ( x ) ~ 0 . There uniquely exists a zo(x,~) for equation (3.9), with 0~< Zo(X,~).~ 0 is a constant. P r o o f The existence and uniqueness of the solution to (3.9) is a simple consequense of the implicit function theorem. From (3.9), we see lira ~,(.:e ,~)----0. If there is a point, say ~0E(0 ,oo ), such that z~ obtains r its negative minmum value at ~0 , then " #~o ~ - I [ ~=~, ~ 0 and "~-~~ l [ ~=~0~0 On the other hand 0:~0 I

----(1-x~)h(x,~/~,~o(X,~/~) = (~ -:~) h. (:~, ~/-T-:~ ,,~ (x, J T : - ~ ) < ( 1 - x z) . / . z , (x,~,.) < 0

+~0(x,~~

+ 0~0 (x ,L) ) ~0(x ,~0)

(0<0<1).

This contradiction shows that ~0 (x ~ ) ~ 0 for all Now, we have

~a.~- ~o - - ~ - - ~ / ( l " x a ) ~ .~0(x,~)

Multiplied by e x p e l / ( 1 - x o

from condition (1.4).

z) l ~-I , the inequality above then becomes

(expE~ ( 1 - x ~ ) l ~ ] . ~ 0 ( x , ~ ) ) < 0

so that o x p E ~ / ( l _ x ~ ) / ~ - i ~o(X,~) - ~ , ( x , 0 )

~ 0. ' Thus

Singularly Perturbed Semilinear Elliptic Equatmn

83

~o (x,~) ~<20 (x,0) e x p [ --~/(1 - - x ' ) l $ ] = a ( x ) e x p [ - - ~/(1 --x2) l ~] It now comes to the conclusion that 2~0(x,~) is of the boundary layer type with respect to ~. The analogous result for a ( x ) < 0 is the same by this way. Assume that all the functions g~ (x, ~) are known now with i-----0,1,2 ,... ,n -- 1 . and they are all of boundary layer type. Our task is to search for 2. (x,~) satisfying 1 1--x z -Oz2" ~=

h

" " ,,kx,M 1--x z ,flo(x, l~-]"-Z'~---x z ) +2o(x,~))~,.+W.(x,~)

~,(x,0)=0

(nf2m--1), ~.(x,0)=--~.,(x,

w/~)

(n----'2m)

(3.11)

~. (x ,~) ~ 0 (as ~ +oo) The assumptions above ensure that, to (3.11), W.(x,~) is ofboundarylayertype. Ifg0(x ' ~) = 0 , boundary problem (3.11) reduces to a second order ordinary differential equation, and then it can be solved with ease. If g0(x,~)~v0 (a(x)~0), we set

O~o fl(x,~)--------~----

J

2 ( 1 , x ~) ;~o h ( x , ~ , f * o ( x , , r 9.

when n=2m--1, --1 + c ~ x ~ l

(3.12)

- - 3 , then

~. ( x , , ) = - - f l ( x , , ) or when n~--2m, --1 + ~ x ~ l

1 - - x- r ) +r)dr

0

! 0r f l -z( x , , ) I : ~ W . ( x , r i ) f l ( x , r l ) d r l d t

(3.13)

--(~ , then

~..(x,~) -----B(x,~) B-~(x,~)

W. (x ,~) B (x ,rl) dndt

B(xA)

(s.x4)

-n..(x, #TZk-r) 9 #(x,0)' and these well describe the asymptotic properties of ~. (x,~) Analogously, we also need to define a boundary layer function

~(x,~, r T ) ~ , ( ~ , ~ ) ~

~

( ~=

y + ~/-Fz~.

)

(3.15)

to portray the boundary layer phenomenon near 0~2----012N Q- . We may repeat the same procedures to determine all functions ~. (n----0,1,9.,...) term by term and we omit the details here.

IV

Main Results

This section is devoted to the establishment of the asymptoticexpansion for problem (1.1}-(1.2) and its existence and uniqueness of the exact solution. We shall also give the distinction. Between the exact solution and its asymptotic expansion an appropriate estimate in the maximum norm. We could remark thal~our asymptotic expansions are only defined on a very small strip along their boundary, respectively. For example, the interior layer funetion V(x, t,,,,/~)isjust defined when t is very small. For this reason, it's necessery to construct a suitable asymptotic expansion for the problem which is uniformly valid in the domain.

84

Zhou Zhe-yan Choosing ,o, to be a small positive number, we then define somesufficiently smooth functions

by

,

r

(o<,
{1,

'<'>--

o p

r

=

{1,

=

o,

(,>COo).

o,

(-+ao
( t < - ~ p,o ) . 1 ( O<~7~-~P, ) , (

9"

Now, for --1+ 3.~x.~
f P,. (x ,v,e) •/7,. (x,y,,) + O'(t) It,, (~ ,t, ~,%')

P . O,,y,~) = / P" (~'v'~) = u .=.(x,v=,~) + ~ (t) P~,. (x, t, ~ - ) +~(,7)z,,.(x,,7,~'7)

_

( (x,v)~.~_).

where m

tZ.(x,y,~) = E ~ . ( x , v ) e . ,

u . . ( x , v , e ) = 5q, n . ( x , v ) ~ ' ,

tl-O

tl-O 2m,

,.

2m

n

V~,. (x,t,~/-F) = ~ ~,. (x.t) ~2, ~ . . (x,t,~/-g-) = E ~. (~, t)e-q n~O

n-O

2m

~

gm

n

z~. (x,~,,,,'-F) = ~ ~(x,~)~ ~. z~. (x,,7, , , / 7 3 : Yq ~(x,,7)~ ~. ~1~0

n~ 0

Clearly, P,,, (x ,y ,e) , defined in such a way, is of course continuously differentiable in ( - 1 + c~ ~ x E 1 - - 3 ) , and twice differentiable, respectively in ~ § and ~ _ We turn now to the consideration about the existence of the solution for problem (1.1)-(1.2). Observe the following problem

,XU=h(x,v,U) U=0

((x,v)EO)

(4.2)

( (x,y)EO~)

where I-2cR' is a bounded domain with # ~ E C r , and suppose h(x, V, U) satisfies: i) h(x, y, U) is of class Cr for any u6.R fixed. ii) #h/igUis continuous and ah/oU>~l>o holds in D. We shall use a fixed point theorem stated below. T h e o r e m 1 (Leray-Schauder) Let E be a Banach space and U=T(U,k)be a functional equation with 0 ~ k ~ l Also suppose i) for any k, T(U, k) is a completely continuous operator in E and uniformly continuous with respect to k in any bounded subset of E. ii) when k---0, equation U-----T(U,o)exists and only one solution exists. (4.3) Morever, operator V=U--T(U,O) is one-to-one in E. iii) all possible solutions of U = T (U,k) are uniformly bounded.

Singularly Perturbed Semilinear Elliptic Equation Then, equation U = T (U,k) has at least one solution for any k. We have the following theorem: T h e o r e m 9. Let condition i ) - iii) in (4.3) be true. Then the boundary value existsand only one solution exists which is of class C Cz+'~(~) . P r o o f We first consider the following problem

AU----kh(x,y,U)

in ~

85

problem (4.2)

(4.4)

UI ~----0. where k is a parameter with 0 ~ k . ~ l . Let E----.Cr ( ~ ) be a Banach space with its norm denoted by I1" II, . Then for every V ( x , y ) E E , there uniquely exists one Solution U , ( x , y ) to the problem

AU~kh(x,y,V)

in O

{ Ul,,~=0 U, ( x , y ) i s of class

(4.a)

CeZ+*~( ~ ) , and the following inequality holds

IIu lh+o
( 4 6)

where the constant c ~ 0 is independent of k and V. Define the operator T(V, k) by

V V (x,y) EE,V (x,y) ~Uk--'--T (V,k). Let S(V)----{V(x,y) I V ( x , y ) E E , IIV ( x , y ) l l . ~ M } and 8 , ( U ) = { U ( x , y ) [U(x, Y) E C Ca+*~ ( , ~ ) , 3 V E S ( V ) s U - - T ( g , k)} . T h e n , from ( 4 . 6 ) , S(v) is a bounded subset in E and so is S , (u) in C r247 ( ~ ) Due to the fact that the space C r can compactly be embedded to the space C o+~ ( ~ ) and also can the space C 0+*~ ( ~ ) , to C (~ (f2) , we are assured that S,(U) is an compact subset of Cr ( ~ ) , and therefore operator T(V, k) is completely continuous. Let U~(x,y)= T(V,kl) and U 2 ( x , y ) ~ T ( V , k 2 ) , for V E S ( V ) , O~.~k~, k~.~l. Then

.f ,x ( u , - u ~ ) -- (k, -t:2)h ( x , y , V ) , t. (U, - U , ) I ~ = 0 . By Schauder type estimate, we have

IIUl - u , II,+~

j k, -;r

gh (x,y,V) I1o.

where the value lib (~,y Y)II o is bounded, uniformly in v. This explicitly shows that T(gk) is continuous inS(V), uniformly in k. We now take condition ii) of theorem into account. When k----0 , (4.4) becomes

AU=o, Ula•----0. For the problem above of course there only exists one solution U - - 0, so that V - T (V ,0) is a unit operator and thereby is one-to-one. To verify condition iii) in theorem I is to construct a prior estimate for all possible solutions for (4.4). The estimates for all possible solutions of (4.4) in maximum norm are uniformly bounded, that

86

Zhou Zhe-yan

is, by maximum principle, 11Uk110~
U , ( X ) = I F , ( X , Y ) H ( Y ) d y +I F i ( X , Y ) H ( Y ) d y where F, (X ,y) =

(4.7)

F2(X,Y)----- 21~ l n l Y J I X - Y I , X,

lnIX-YI,

YED and

H (g) --h (g,u~(Y)) Denoting D -i - - ~ , 0

c) D2----- ~y

, we also have

D,U,(X)=I D , I " t ( X , Y ) H ( Y ) d Y + I D,F~(X,Y)H(Y)dY-----I,+I,

(4.8)

Because U, is uniformly bounded, there exists a constant M~ >0 such that the estimate for H(X) holds

11H(X) I1~ IIh (X, U, (X)) U0~M,

(a. 9)

uniformly in k with 0 ~ k ~ ! . ~<: 1 f We first look at [ I, [ .~.-2--z--j~ ] D~I n [ X - Y [ [ ] H (Y) !dY

MI "~ 2z jf~ I D J n [ X - Y I [dY,

Let eo ~ 0 be a constant, and ~eo = {Y[ [: l ' r - X [ ~eo} , X = (x~ ,x~), Y = (yt ,Yz) 9Then. by using the following inequality

D ~ l n l X _ Y l = x~-y~ IX--YI ~ we therefore

get[ [ t i e

2~r'

]X_YlZD-Qeo

When

YE~-~eo,

IX-Yl~>~e~o

(4.10)

[X_YlZ ~

--~

~ I +1.

Qeo ,

then

MI zze0 j~

D-Deo

Let Yi ----pcos8 + xx,

3el

(4.11)

y~=psin8 + x~. This immediatelycomes tothe result M I

gw

Thus (4.13) Analogous estimate for part 12 in (4.8) is really in the samt way~ By virtue of theorem 1, however, the boundary value problem (4.4) has a solution for all k(0-.~.~k~l) especially for k-- 1. Let U(X) be such a solution with k = 1 , Then U (X) @E and by Schander type estimate therefore U(X)6.C (2+')( ~ ) .The uniqueness for the solution to

Singularly Perturbed Semilinear Elliptic Equation

|

87

exist is just a direct consequence of the maximum principle so that the proof for theorem 2 is completed. Theorem 2 shows that the singular perturbation problem (1.1)- (.1.2) has a unique solution

U. ( X) ECC~*"~(ga ) . Putting Z~ (x,y) =U, (x ,y) - P . , (x,y)

, we shall prove that Z..(x, y) is of 0 (era+89~as

e tends to zero, namely

IlZ=(x,y) I]o---O(e "*89 ), as (e~O) .

(4.14)

D e f i n i t i o n Suppose that, to the problem

AUfh(x,y,U), ( ( x , y ) E O ) , { U=~(x,y), ((X,y)E~). there exist the functions a ( x , y ) inequalities

(4.15)

and fl (x ,y)to be of class Co~ ( ~ ) , which satisfy the following

A_a(x,y) ffi l i m [ i n f 5r~-~ OL

Oa(x+T,y) Oa(x - T , y ) Ox ar Ox 2T

O a ( x , y + T)

Oa(x,y-T)

Oy

+ inf

2T

Oy

]>~h(x,y,a(x,y) ) , (x,u)~

A.,0(x,y) ~ l i m ~ s u p

+ sup

(4.1e)

Ox

Ox

2T

o,O(x,ll+ T) Oy

O~O(x,y.T) Oy 2]"

]
andon 0 ~ , a(x,.y)<9(x,y) ~ fl(x,y) ,then a(x,y) and fl(x,y) super-and subsolutions to (4.15), respectively. We difine for problem (1.1)-(1.2) the function by

p(x ,y,e) ---P.. (x,v,e) +~:e'*89 where c, a constant, will be determined lately. P,m (x ,y ,e) is twice differentiable in D+ and l~_, respectively, and thus or ~ _ . This reduces (4.17) to

eAp-h(x,y,#(x,ia) ) <~o s

9

aO(x-T,y~

o# (x + T ,y)

t4.tT) are said to be the

(4.18) eA+=~A in ~+ (4.19)

By (4.1) and a simple calculation, inequality (4.19) holds for fl (x ,U ,e) defined by (4.18) in and ~_ with --1 + c ~ x ~ . l --6 9 O n / " ; we note that P,, (x ,y ,e)is of class CcZ~( ~ + ) and C cz~( ~ .) , and then we have.

A +P., (x,y,e) = 1[ 2 [LXP,.+ A.~,,]

(4.20)

88

Zhou Zhe-yan

and since P,,EC 0~ ( ~ ) , by (4.20), also

eA+#(x,O,e) --h (x,O,#(x,O,e)) 1

------~-EeAP., (x ,0, e) - h (x,O, P., (x,O, e) + e~'*89 ) ]

+-I

[eaP'(x'O'e)--h(x'O'P"(x'O'e)+ce"+ 89

= I + I

(4.2J)

Here, we do use the properties of the function P= to describe operator A +byLaplace operator A in the neighhourhood of the/". So it is easy to verify that I ~<0 and I ~<0 in (4.21), provided we choose c in (4.16) to be large enough. The same calculation leads us to the result that inequality (4.16) also holds in O with--.1 + ~ < ~ x ~ --c3 f o r a ( x ,y,e), instead of f l ( x , v , e ) , defined by

a ( x , v , ~) : P , . (x ,v ,e) - ee"+ 89

(4.22)

if we let c be large enough. L e m m a 4.2 The solution of (1.1)-(1.2) is uniformly bounded in e . P r o o f Using the maximum principle we learn that there are constants m and M such that m ~ U , (x,y) ~
m:min{min{Uo(x,y)

}, min.[~o(x,v) }~ and

(e,IOEQ

(r

EOD

M f f i m a x ~ m a x { U o ( x , y ) }, m a x ~ ( x , v ) C8~10 EO

}}.

(Btl O EOD

Clearly, m and M are both independent of e .Then our proof ends. Let N be a large enough constant such that N>~rnax.l I m l , IMI } , and IU,(-l+6,y) tU,(t-3,y)

I~
(4.23)

tbr ( 1 - 3 , y ) E ~

(4.24)

to construct the function F (x,e) as the followin~ form

]+

,].

with the properties stated in a lemma below: L e m m a 3 i) F(x, e )>0, for - - l + ~ < x ~ < l - - c S !

dtF

, r.

ii) F(x, e ) satisfies the equation e--~-r--~, r =O;

;

iii)F(x, e ) = O ( e n ), as e--)0, - - 1 + 2 c 3 ~ x < t - - 2 J , constant. Now, we set

where n is an arbitrary positive

/~ (x,y,e)

:/~ (x,g,e) + F (x,e) (x,v ,e) ---a(x ,v',e) - F (x,e)

(4.25) (4.26)

Then, we prove, due to (4.16), that for - - l + ~ < x - ~ l - - r

eA,~ -h(x,y,~

d~F ) ~e~-h,,(x,y,#+OF)

<~e~ffr-IF=O .

F

(0<0<1)

(4.27)

Singularly Perturbed. Semilinear Elliptic Equation

89

and also that

eA_S (x ,y ,e) - b (x ,y, ~ (z W ,e) ) >~0

(4.28)

On 0g2, ( -- 1 + 3 ~ < x ~ l -- 6), however,

(x,y,e) =P,,,+ce'+4z-+F(x,e) >ep(x,y) = U , (x.,V) ~(x,y,e) =P,~--ce~+@ - F ( x , e ) < q ~ ( x , y ) f U o ( x , y )

and

(4.29) (4.30)

Furthermore, on x = - - 1 + 3 ,

B (x,y,e) >~N>~U. (x ,y)

(4. Sl)

~ (x ,y,e) ~ - N ~ U ~ (x ,y)

and

(4.32)

for e is sufficiently small. Inequalities (4.31) and (4.32) also hold for x----1--6. Combining those with the definition of sub-and supsolution we have proven: L e m m a 4 /g (x,y,e) and ~t(x,y,e) , formulated by (4.25), (4.26) respectively, are the sub-and supsolutions to the boundary problem (1. I)-(1.2) on ~ with --1 + 3 ~ x ~ 1 - - 6 We also need the following result: L e m m a 5 If there exist the supersolution a (x, ~) and subsolution fl ( x , y ) to the problem AUfh(x,y,u) in l'~

{U=cp(x,y),

in 092

then any solution U(x, y) of the problem satisfies the ineguality

a(x,y) <~U(x,v)
A_ (U-fl) =AU (Xo,yo~ -A+fl(Xo,y~ >~h(xo,yo,U (xo,yo) ) -h(xo,yo,fl (Xo,Vo) ) = h, (xo ,y~

9[-U (xa,y~ - f l (xo,y~ ] ~ l . E U - f l ] > 0 ,

a contradiction; this proves the Lemma. From Lemma 5, we see that there is a small enough constant eo~>0, When 0 < e < e 0 -- 1 + 3 < x < 1 --6 , such that

(x,y,e) ~
IZ . (x,v,e) I -- l U, (x,y) - P , , (x W,e) I< F (x,e) + ce" +'t'. But for - - l + 2 3 < x ~ < l - - 2 d , F(x,e) is of O(em*'~) . Thus

IIZ,,(x,v,e) IIo=O(e "+ 89

,

(x,y)6~,(-]+23<~x~l-23)

(4.33)

3 If conditions (1.3) and (1.4) are satisfied, there exists a unique solution U, (x, y ) EC (z*~~( 5 ) to the perturbation problem ( 1.1)- (1.2) for 0 < e ~ e 0 and estimate (4.3 3) for the remainder Z,, (x , y , e) holds. Theorem

References

[ 1 ] Jiang Fu-ru, Some questions on the turning point theory in ordinary differential equations,.An

Academic Thesis~Collectionfor the Conference of Second National Modern Mathematics and

90

Zhou Zhe-yan

Mechanics, Shanghai, China, (1987). [ 2 ] Howes, F.A., Boundary-interior layer interactions in nonlinear singular perturbation theory, Amer. Math. Soc. Memoirs, 208 (1978). [ 3 ] Howes, F.A., Differential inequalities of higher order and the asymptotic solution of nonlinear boundary value problems, SIAM.J. Math. Anul., 13, 1 (1982). [ 4 ] Chang, K.W. and F.A.Howes, Nonlinear perturbution phenomena: Theory and application, Appl. Math. Sci., 56,Springer-Verlag, New York Inc. (1984). [ 5 ] Howes, F.A., Some singularly perturbed superquadratic boundary value problems whose solutions exhibit boundary and shock layer behavior, Nonlinear Analysis, 4 (1980), 683 -698. [ 6 ] Dejager, E.M., singular elliptic perturbations of vanishing first-order differential operatolrs, Lecture Notes in Math., 280, Springer-Verlag (1972), 73-86. [ 7 ] Howes, F.A., Singularly perturbed semilinear elliptic boundary value problems, Partial Differential Equations, 4, 1 (i979), 1-39.