International Journal of Modern Nonlinear Theory and Application, 2013, 2, 170-175
http://dx.doi.org/10.4236/ijmnta.2013.23024 Published Online September 2013 (http://www.scirp.org/journal/ijmnta)
General Boundary Value Problems for Nonlinear
Uniformly Elliptic Equations in Multiply
Connected Infinite Domains
Guochun Wen1, Yanhui Zhang2, Dechang Chen3
1School of Mathematical Sciences, Peking University, Beijing, China
2Mathematic Department, Beijing Technology and Business University, Beijing, China
3Uniformed Services University of the Health Sciences, Bethesda, USA
Email: wengc@math.pku.edu.cn, zhangyhchengd@yahoo.com.cn, dechang.chen@usuhs.edu
Received May 1, 2013; revised June 29, 2013; accepted July 3, 2013
Copyright © 2013 Guochun Wen et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This article discusses the g eneral boundary value problem for the nonlinear uniformly elliptic equation of second order

,, ,,,in,
zzz zzz
uFzuuuGzuu D (0.1) and the boundary condition
 
12
22on
uczu cz
,
(0.2) in a
multiply connected infinite domain with the boundary
D
. The above boundary value problem is called Problem G.
Problem G extends the work [8] in which the equation (0.1) includes a nonlinear lower term and the boundary condition
(0.2) is more general. If the complex equation (0.1) and the boundary condition (0.2) meet certain assumptions, some
solvability results fo r Problem G can be obtained. By using reduction to absurdity, we first d iscuss a priori estimates of
solutions and solvability for a modified problem. Then we present results on solvability of Problem G.
Keywords: General Boundary Value Problems; Nonlinear Elliptic Equations; Multiply Co nnected Infinite Domains
1. Formulation of Elliptic Equations and
Boundary Value Problems
Let be an -connected domain which in-
cludes the infinite point and has the boundary
D
1N
0
N
j
j
 
in , where

201C
.
Without loss of generality, we assume that D is a
circular domain in 1z, where the boundary consists
of circles
1N
01 1
Nz
 ,

,1,,
jjj
zzr jN   and . Note zD
that this article uses the same notations as in references
[1-8]. We consider the nonlinear uniformly elliptic
equation of second order


 

123
,, ,,,,
Re ,
,,,,, ,,
,, ,1,2,3.
zzz zzz
zz z
zz
jj z
uFzuuuGzuu
FQuAuAuA
GGzuuQQzuuu
AAzuuj




zz
(1.1)
This is the complex form of the nonlinear real equation

,,,,,,,0
x y xx xyyy
xyuu u uuu (1.2)
with certain conditions (see [3]). We suppose that the
Equation (1.1) satisfies Condition C, as described below.
Condition C 1)
 
,,, ,,,1,2,3
j
QzuwUAzuwj
zD
are measurable in
for all continuous functions
,uz wz in D and all measurable functions
0,2 ,
p
Uz LD and
satisfy

 
,2 10,220
,231 2
,, ,,,, ,,
,,,,,,0in,
pp
p
LAzuwDkLAzuwD k
LAzuwDkAzuw D






(1.3)
in which

0001
,2,,,1ppppkk

are non-ne-
gative constants.
2) The above functions are continuous in
,uw
 , for almost every and ,zDU
1,2,3j0, 0
j
QA for .zD
3) The Equation (1.1) satisfies the uniform ellipticity
condition
1201
,,, ,,,,
2
F
zuwUF zuwUq UU (1.4)
for almost every point ,zD
any functions
,uz wzCD and where
12
,,UU
01q
C
opyright © 2013 SciRes. IJMNTA
G. C. WEN ET AL. 171
is a non-negative constant.
4) The function possesses the form
,,Gzuw

12
,,in ,Gzuw BwBuD

 (1.5)
where are continuous functions in
 
,uz wzD,
,2
,
pj
LB
k0 0
0,, 1,2,2Dkj pp


  
 for a
positive constant .
0
According to [7], we introduce the general boundary
value problem for the Equation (1.1) in D as follows.
Problem G Find a continuously differentiable solution
of the second order Equation (1.1) in

uz D
satisfying the boundary conditions
 
  
12
12
22,
..Re, .
z
uczu cz
iez uczuczz


 

(1.6)
Here
is a given unit vector at the point ,z
and
cos,cos,,zxi
 
y
z
and are real
functions. We assume

z
1
,c
and satisfy the con-
ditions 2
c
01 02
,,,,,CkCckCc
 

2
,k

(1.7)
and
 
1cos ,0,,czn z

in which

02
are non-negative con-
stant, and is the unit outer normal at
121, ,,kk


nz
. If
then we assu-
me that
 
1
cos,0 on,1,
j
z jN
0,nc


***
2
d0,1, ,1
jjjj
czzuabbkjN

2
, (1.8)
in which *
j
a is a point on
j

,N
and are
real constants. There is no harm in assuming that
on
*1, ,
j
bj N

,n
 
1
cos ,0ncz

*
 
0,
0
10
NN and cos
1z
.
c
do not both vanish identically on 0
** 1
N
N
 
We can see that the above bound ar y cond itio ns in clu d e
some irregular oblique derivative boundary conditions. If
on , then Problem G is the regular
oblique derivative problem (Problem III). If
and 1 on

cos ,0n

cos ,0n
c0
, then Problem G is the
first bound ary valu e problem, i.e., the Dirichlet boundary
value problem (Problem D), in which the boundary
condition is
 


****
2
1d,1,1,,
j
z
jjj
a
uz rz
czsbrabjN

1.
(1.9)
One problem regarding the well posed-ness of
Problem G for (1.1) can be formulated as follows:
Problem H Find a system of continuous functions
of the equation
 
,uz wz
 

 

123
,,,,, ,
Re ,
,, ,,,,,
,, ,1,2,3,,
zz
z
z
jj z
wFzuwwGzuw
FQwAwAuA
GGzuwQQzuww
A
Azuwj wu




(1.10)
satisfying the modified boundary conditions

  
12
12
22 ,
..Re, ,
z
uczucz hz
iez uczuczhzz






(1.11)
and the point conditions:

000
1,0,1,,,, 1,,
jj j
uabjmaa ajm .
(1.12)
An explanation of the above conditions is given as
follows. The bound ary
can be divided into two parts:
1
cos ,0,0ncz
 and
cos,0,0 ,ncz
 
1 such that

11
,,
,,,,, ,
ml
Ea aaa
 
  




every component of
and
includes its initial point, but does not in-
clude the terminal point, and there is at least one point on
each component of so that
 ,
cos ,0.n
The
points
m
j
a
1, ,
j
aj and j possess the
following property.

1,, laj
and j, when the
direction of a

at ,
j
j
aa
is the same as the direction of
.
j
a
and j
a
, when the direction of
at
,
j
j
aa
is opposite to the direction of . And (cos )n,
changes the sign once on the two components of ,
with the end point j or j. And aa
0, 1,bj,
j
in (1.12) are real constants satisfying the condition:
m
3j herein 3 is a non-negative constant. More-
over, the undetermined function in (1.11) can be
written as
,
kkb
hz
,,0,1,
jj j
hzhzzjl

,. (1.13)
In (1.13)
*0,1, ,
jj j
 l are non-degen-
erate, multiply disjointed arcs, each of which consists of
inner point s of
0,1, ,
jjl
, such that
0, 0z

cos ,n
on
000
, ,,.jla E

1,
j

In addition,
,l

jz
j
0, 1,hj
j are unknown real constants to be
determined appropriately, and is a positive
function on
and
0
jz
on
j
 and
Cz
0j
 ,,0,1,,,kjl

 in which
12 1

and 0
k are non-negative constants. It is
not difficult to see that the index of Problem H is given
by

1arg1 .
22
ml
KzN
 
(1.14)
Copyright © 2013 SciRes. IJMNTA
G. C. WEN ET AL.
172
If on , then
In this case, Problem H for (1.1)
is called Problem O or Problem IV, which includes the
Dirichlet problem, the Neumann problem and the regular
oblique derivative problem as its special cases. We note
that except the case where and
 
1
cos ,0,0ncz

,,.E

 
cos

,n
 
0
10cz
on , the conditions (1.12) and (1.13) can be replaced
by

 
0
1,0,1,,,
,,1,
jj
jj
ua bjm
hzhzzjl


,.
(1.15)
with
3,0,1,,
j
bkj m, (1.16)
in which 3 is a non-negative constant. Also note that
[4,7] discuss the corresponding problem for the equation
(1.1) with in the bounded dom ains.
k

,, 0
z
Gzuu
2. A Priori Estimates of Solu t io n s o f
Boundary Value Problems
We first give a priori estimates of solutions of Problem H .
Theorem 2.1 Suppose the second order nonlinear
Equation (1.10) satisfies Condition C, and
in (1.3),
(1.7) is small enough. Then any solution
 
,,
z
uzwzuz u


,, 0Gzuw
of Problem H for (1.10) with
satisfies the estimates
 
0
1,2 1
,,,
pzzzz
SuCuz DLuuDM
 

 
(2.1)

2*2123,SuMkM kkk 
in which

0
min,1 2,p


0
2,pp

1100
,,,,,,
M
Mqpk KD

22000
,,,,,

123
,,kkkk,
.
M
Mqpk KD
Proof First of all , we prove that the solution
uz of
Problem H satisfies the estimate


1
13300
,,,,,,SCuzD MMqpkKD


 .
Suppose that the estimate (2.3) is not true. Then there
exist sequences of coefficients




*
123 12
,,,,,,,,
nnnn
nn njnjn
QAAAcc bb
*
123 12
,,,,,,,,
of
(1.10), (1.11), (1.12) and (1.15) satisfying the same
conditions of
j
j
QA AAc cbb

,
n
Q

1,
n
A

2,
n
A

3
n, such that
A
in weakly converge D
0000
,,,QAAA
to respectively, and
123
 

*
12
,,,
njn
ccb
,
nn
j
n
b
*
,,
on uniformly
converge to 01
020 00
,,
j
j
bbcc
respectively, and the
corresponding bou nda ry valu e problems
1232
Re,0 in,
nn nnn
zzzz z
uQuAuAuAA


 D (2.4)


12
1
222,
cos,0on,d0,
j
nnn
n
nn n
ucu ch
cz ncs



** 0
1,1,,,
1,0,1,,,1, 2,
jjn
jjn
ua bjN
ua bjmn

 
(2.6)
have the continuously differentiable solutions
1, 2,
n
uzn with the property that
1,
nn
HCuD



as There is no harm in .n
assuming that Denote
1,1,2,
n
Hn
,1,2,n
nnn
UuH
It is clear that the function
n
wz Unz
is a solution of the following Rie-
mann-Hilbert boundary value problem
11
Re,in ,
nn nnnn
nznz nn
wQwAwAAAuA

 
 3
D
(2.7)

** 0
1,1,,,
1,0,1,,,
nj jn
nj jn
uabjN
ua bjmn

 
1,2,
(2.8)
where the index of
z
n is
12KN ml,
and
,1
n
Cw z D

showing that
n on
wz D is
bounded. According to the method in the proof of
Theorem 4.7, Chapter I [4], we can obtain that
n
wz
satisfies the estimate
0,2 4
,,
nnpnznz
LwCw DLwwDM
 

 ,
(2.9)
in which
4400
,,,,,,
M
Mqpk KD
and then
  
2
(2.5)
*0
2
1
2Re d
j
zn
nn
a
wz
Uz zuzH
z
 
satisfies
0
1,2 5
,,
nnpnzznzz
SUCU DLUUDM
 

 ,
(2.10)
where
5500
,,,,,.
M
Mqpk KD
Hence from
n
Uz and
nz
U, we can choose the subsequences

k
n
Uz and
k
nz
U, which uniformly converge to
0 and 0
Uz
z
U in D respectively, such that
0
is a solution of the following boundary value problem
Uz
00 00
122
Re0,0in,
zzzz z
UQUAUAUA


 D (2.11)


100 100
0
22,cos,0 on,
Ucuhczn

(2.12)

*0
10,1,,,10,0,1,, .
jj
Ua jNUajm 
(2.13)
By the uniqueness of solutions of Problem H (see
Theorem 2.3 below), we see that
0Uz on D.
However from
1,1,CU zD

n

it can be derived that
10.,1CUzD


This contradiction proves that (2.3)
is true. Afterwards, using the method of deriving (2.9)
from 1,1,CUD

n

we can obtain the estimate (2.1).
The estimate (2.2) can be concluded from (2.1).
Theorem 2.2 Let the Equation (1.1) satisfy Condition
Copyright © 2013 SciRes. IJMNTA
G. C. WEN ET AL.
Copyright © 2013 SciRes. IJMNTA
173
dary conditions: C and
in (1.3), (1.7) be a sufficiently small positive
constant. Then any solution
 
,wz uz
of Problem
H for (1.10) satisfies the estimates
123
Re, ,
zz
wQwAwAuAGzD
(2.16)
 
12
Re, ,zwzcuczh zz

 
 (2.17)
 
6*
,,C wzDCuzDMk




, (2.14)

** 0
1,1,,,
1,0,1,,,1, 2,
jjn
jjn
ua bjN
ua bjmn

 
(2.18)
00
,2,27 *
,,
pzzpz
LwwDLuDMk

 

, (2.15)
where 0
,p
are as stated in Theorem 2.1,

000
,,,,, ,6,7,MMpk KDj

jj
q
 
*1230 ,,
kkkkkCwD CuD


 

.
By using the same method as in the proof of Theorem
2.1, we can obtain the estimates (2.14) and (2.15).
Now we discuss the uniqueness of solutions of
Problem H for the nonlinear elliptic Equation (1.1) with
,, 0Gzuw
. For this, we need to consider the follo-
wing condition
Proof It is easy to see that
 
,wz uz
of Problem
H for (1.10) satisfies the following equation and boun-
 

0
112211 221 2
12,20 0
,, ,,,,Re,
,,,,1,2,,,2,
zz z
jjp j
F
zu uUFzuuUAuuAuu
AAzuuUjL ADkpp







 
(2.19)
for any continuously differentiable functions


1,1,
j
uz CDj
2 and any measurable function


0,2 ,
p
Uz LD where

0
min,1 2,p


H for (1.10). By the above conditions, we see that
12
uzuzuz is a solution of the following
boundary value problem Problem
12
Re0,,
zzzzz
uQuAuAuz



 
2ppp
,k
00 0 are constants as stated in Section 1.
We can prove the uniqueness of solutions of Problem H
for (1.1).
D (2.20)
 
1
22,
uczuzHz z
,
 (2.21)
Theorem 2.3 Let the second order nonlinear Equation
(1.1) satisfy Condition C and (2.19) with 2 in .
Then the solution of Problem H for (1.10) with
0A
D
,, 0
z
Gzuu is unique.

*0
10,1, ,, 10,0,1, ,,
jj
uajNua hm 
(2.22)
Proof Let be two solutions of Problem
 
12
,uzuz with
 
 

 
 
0
12111 112
11211 2122
12 222 212
12
2
12
0,2 2
Re, ,,, ,,,
Re,,,,,,,
,, ,,,,for ,
0for
1,,,1,2,0in
zzzz zz
zz
zzz zzz
z
zzz zzz
pj
QuuFzuu uFzuu u
AuuFzuuuFzuuu
Fzuu uFzuuuuz uz
uu
A
uzuzz D
QqL ADjA

 


 




 

 
,D
,,
,,GGzuzwz
is coninuous and bounded with
where 001
are non-negative constants. Accor-
ding to the proof of Theorem 2.6, Chapter I, [4], and
using the extremum principle of solutions for (2.20) (see
Chapter 3, [3]), we can prove that in , and
then in .
,,qpk
 
1
uz u

0uzD
2
zD
 
 
,2,2 1
,2 2
,, ,,
,,
pp
p
LGzuzwzDLBD
CwDLBDCuD,,








(3.1)
3. Solvability of Boundary Value Problems
We first prove a lemm a.
Lemma 3.1. If satisfies the condition
stated in Condition then the nonlinear mapping :
,,Gzuw
,C T
  
,2p
CD CDLD defined by
where 02.pp
Proof In order to prove that the mapping
T
:
,2p
CD CDLD defined by
,,GGzuzwz
 
is continuous, we choose any
sequence of functions


,0,1,2,
nn
uz CDn,,
nn
wzuz wz

G. C. WEN ET AL.
174
such that 00
,,
nn
CwwDCuu D



0
as
Similarly to Lemma 2.2.1 [5], we can prove that
possesses the property
that
.n
C G

00
,, ,,
nnn
zuwG zuw
,2 ,0as
pn
LCD n


 . (3.2)
And the inequality (3.1) is obviously true.
Theorem 3.2. Let the complex Equation (1.1) satisfy
Condition C, and the positive constant
in (1.3) and
(1.7) be small enough.
1) When 0,1
1,2
,p
wz uzWD0 with the constant
00
2ppp as stated before.
2) When
min ,

1, Problem H for (1.10) has a
solution
,wz uz
, where

0
1,2,
p
wz WD pro-
vided that

0
8,232 0
,,
m
p
j
j
M
LADCcb


 (3.3)
is sufficiently small.
Proof 1) In this case, the algebraic equation for
t
becomes
,
 Problem H for the Equation
(1.10) has a solution

,uz


,wz where

9,23,21,2220
,,, ,
m
pp pj
j
,
M
LADLBDtLBDtLcb t


 


 

(3.4)
with 967
M
MM, where 67
,
M
M0 are constants as
stated in (2.14) and (2.15). Because ,1,

0. Equa-
tion (3.4) has a unique solution 10 Now we
introduce a bounded, closed and convex subset
tM*
B
of the Banach space
 
,CDCD
whose elements
are of the form satisfying the condition
 
,wz uz
 

 
10
,,,,.wzuzCDCwz DCuz DM



(3.5)
We choose a pair of fu nctions
 
*
,wz uzB


 and
substitute it into the appropriate positions of

, ,,,
z
,,
F
zuwwGzuw in (1.10) and the boundary
condition (1.11) to obtain

,,,,,,,,
zz
wFzuwuwwGzuw
 
(3.6)

12
Re, ,zwzc zuczz

 
 (3.7)
where


1
23
,,,,,
Re,,,, ,
,,,,.
z
zz
Fzuwuww
QzuwwwAzuww
AzuwuAzuw



 
 
In accordance with the method in the proof of The-
orem 1.2.5 [5], we can prove that the boundary value
problem (3.6), (3.7) and (1.15) has a unique solution
,wz uz
. Denote by

,,wuTw zuz

 the
mapping from
,wz uz
 to Noting
that
 
,wz uz
.
,2210 0110 0
,,,
p
LAuD MkCcuMk



 ,
provided that the positive number
is sufficiently
small, and noting that the coefficients of complex Equa-
tion (3.6) satisfy the same conditions as in Condition C,
from Theorem 2.2, we can obtain



00
,2,29,2 32,2
0
98 ,21,2298 ,2110 ,221010
,,,,,,,
,, ,,,,
m
pzzpzpjp
j
pp pp
.
wDLwwDCuDLuD MLADCcb L GDC
M
ML BDCwDL BDCuDMMLBDML BDMM
 

  
 

  

 
 
 

(3.8)
This shows that
T
maps onto a compact subset
in Next, we verify that
*
B
.
*
B
T
in is a continuous
operator. In fact, we arbitrarily select a sequence
in such that
*
B

,
n
wz

n
uz

*
B,

00
,,0as
nn .w DCuu Dn
 
Cw (3.9)
By Lemma 3.1, we can see that


,20 0
,,,, ,
01,2,3as .
pjnnj
LAzuwA zuwD
jn


 
 (3.10)
Moreover, from
00 00
,,,,,
nn nn
wuTwu wu Twu
 
, it is clear that
0nn0
,wwuu is a solution of Problem H for the
following equation:


0
00000
00
,,,,,
,,,,,
,,,, in,
n nnnnnz
z
z
nn
ww Fzuwuww
Fzuwuww
Gzu wGzuwD




 
(3.11)





0
10
Re
on ,
n
n
zw w
czuuhz




(3.12)
Copyright © 2013 SciRes. IJMNTA
G. C. WEN ET AL. 175


**
0
0
110,1,,
110,0,1,, .
nj j
nj j
uaua jN
uaua jm


0
,
(3.13) In accordance with the method in proof of Theorem
2.2, we can obtain the estimate







0
0
0,20 0
0,20
11,22200 0
,2330 0
,20010
,,
,,
,,,,,
,,,,,
,,,,,, ,
npnn
zz
npn
z
pnnn
pnn
pnn n
Cw wDLw ww wD
Cuu DLuuD
MLAzuwuAzuwuD
LAzuwAzuwD
LGzuwGzuwDCczuu


 




 











  
 

(3.14)
in which

111100 0
,,,,, .
M
Mqpk KD
From (3.9),
(3.10) and the above estimate, we obtain
00
,,0
.n
nn


as On the
basis of the Schauder fixed-point theorem, there exists a
function
CwwDCuu D


  

,,wz uzwzuzCD

 
,,.zT wzuz


such that

wz u And from Theorem 2.2,
it is easy to see that
 
1,2
,
p
wz uzWD
0, 1.
0 and
is a solution of Problem H for the
Equation (1.10) with the condition
,
 
,wz uz

In addition, using a method similar to the above, we
see that if

12
,, ReGzuwBw Bu
 in , where D
,2 0
01,,,1,
pj
LBDk j

 
 2, then the above
solvability result still holds.
2) Secondly, we discuss the case, where

min ,1.

tM In this case, (3.4) has the solution
10 provided that 8
M
in (3.3) is small enough. We
consider a closed and convex subset in the Banach
space *
B
 
,CDCD i.e.,
 

*10
,,,,
BwzuzCDCwDCuDM

 
 .
Applying a similar method as before, we can verify
that there exists a solution
00
of Problem H for
(1.10) with the condition

11
,2 ,2
,pp
wz uzWDWD

min ,


1.
Moreover, if 12
,, ReGzuwBw Bu
 in ,
where D
1,
 ,2 0
,,1,2,
pj
 then
under the same condition, we can derive the above
solvability result by a similar method.
LBDk j


From the above theorem, the next result can be
derived.
Theorem 3.3 Under the same conditions as in The-
orem , Problem G has
3.2 1
l

uz solvability conditions,
and the general solution includes arbitrary
real constants. 1m
Proof Let the solution

,wz uz

0, z
l
of Problem
for (1.10) be substituted into the boundary condition
(1.11). If t he function , i.e.
then we have
H
hz
, ,0,, 0,1,
j
hz j


z
wz u
in
and the function is just a solution of Problem
G for (1.1). Hence the total number of above
equalities is just the number of solvability conditions of
Problem .
D
uz
1l
G
Also note that the real constants
0, 1,,
j
bj m in
(1.12) and (1.15) are arbitrarily chosen. This shows that
the general solution of Problem G for (1.1) includes the
1
m arbitrary real constants as stated in the theorem.
Note: The opinions expressed herein are those of the
authors and do not necessarily represent those of the
Uniformed Services University of the Health Sciences
and the Department of Defense.
REFERENCES
[1] I. N. Vekua, “Generalized Analytic Functions,” Pergamon,
Oxford, 1962.
[2] G. C. Wen, “Linear and Nonlinear Elliptic Complex
Equations,” Shanghai Scientific and Technical Publishers,
Shanghai, 1986. (in Chinese)
[3] G. C. Wen, “Conformal Mappings and Boundary Value
Problems,” American Mathematical Society, Providence,
1992.
[4] G. C. Wen, “Approximate Methods and Numerical
Analysis for Elliptic Complex Equations,” Gordon and
Breach, Amsterdam, 1999.
[5] G. C. Wen, D. C. Chen and Z. L. Xu, “Nonlinear Com-
plex Analysis and its Applications, Mathematics Mono-
graph Series 12,” Science Press, Beijing, 2008.
[6] G. C. Wen, “Recent Progress in Theory and Applications
of Modern Complex Analysis,” Science Press, Beijing,
2010.
[7] G. C. Wen and C. C. Yang, “On General Boundary Value
Problems for Nonlinear Elliptic Equations of Second Or-
der in a Multiply Connected Domain,” Acta Applicandae
Mathematicae, Vol. 43 No. 2, 1996, pp. 169-189.
doi:10.1007/BF00047923
[8] G. C. Wen, “Irregular Oblique Derivative Problems for
Second Order Nonlinear Elliptic Equations on Infinite
Domains,” Electronic Journal of Differential Equations,
Vol. 2012, No. 142, 2012, pp. 1-8.
Copyright © 2013 SciRes. IJMNTA