Advances in Pure Mathematics, 2011, 1, 340-344
doi:10.4236/apm.2011.16061 Published Online November 2011 (http://www.SciRP.org/journal/apm)
Some Steiner Symmetry Results in Overdetermined
Boundary Va lue Pr oblem*
Zhongbo Fang, Anna Wang
School of Mathemat i cal Sci ences, Ocean University of China, Qingdao, Chi na
Received June 17, 2011; revised August 2, 2011; accepted August 10, 2011
Abstract
In this paper, we use the moving planes method to prove that the domain Ω and the solution u are Steiner
symmetric if u is a positive solution of the overdetermined boundary value problem in Ω.
Keywords: Steiner Symmetry, Moving Planes Method, Overdetermined Boundary Value Problem
1. Introduction
We will present some symmetry results in overdeter-
mined boundary value problem

,,0 in ufxuu 
0 on u

on
ucx




where

1,n
x
xx R

,


23
,,,
n
x
xx x
 and

is
the outward unit normal to . Here is a bounded
domain of class in ,
 
n
R
2
C
f
is a function of class
and satisfies certain conditions, is a differenti-
able function. Obviously the constant normal derivative
of on is a special case of this problem.
1
Cc
u
For the motion of a viscous incompressible fluid
moving in straight parallel streamlines through a pipe
with planar section or the torsion of a solid straight
bar of given cross section , both of them can be de-
scribed by the following overdetermined problem


1 in u
0 on u
on
uconst



where is a bounded domain of class in
such that there exists a function satisfying the above
problem, and
2
Cn
R
u

is the unit normal to . Using the
moving planes method, J. Serrin proved that if the above
problem has a solution in

u

2

C, then must be
a ball. For more detail, see [1]. Later there are many au-
thors who investigated the overdetermined the problem.
B. Gidas, W. M. Ni and L. Nirenberg proved symmetry
of positive solutions of elliptic equations under the
Dirichlet boundary condition via moving planes method,


in ufu

 
0 on u


See [2]. The paper [3] extended the result of [2], but it
did not contain the gradient. A. Colesanti considered the
positive solutions of a more general p-Laplacian equation
under the overdetermined conditions which has only one
critical point via moving planes method too,





2,0 \ p
divuuf uuinP


0 on u


0 on
uc




.
See [4]. Recently there have been found several other
new approaches in studying symmetry problem, such as
continuous Steiner symmetrization, domain derivatives
and some geometry method, see [5-9]. The moving planes
method is a classical technique and is very useful in
dealing with symmetry problems, so we still use it to
extend the results of [2,3] to the overdetermined bound-
ary problem containing gradient.
We shall prove that

and are Steiner symmet-
ric with respect to the hyperplane

1
u

x

 for some
real number

. To be more specifically, a set
A
is
Steiner symmetric with respect to the hyperplane


1
x

 if it is symmetric with respect to the hyper-
plane


1
x

 and convex in the 1
x
direction. The
*The Project-sponsored by SRF for ROCS, SEM.

Z. B. FANG ET AL.341

function is called Steiner symmetric with respect to
the hyperplane
u

1
x

, we mean that the level sets of
are Steiner symmetric with respect to the hyperplane
1
ux


. Throughout this paper, we consider

as a
bounded domain of class in unless otherwise
stated.
2
Cn
R
Section 2 of this paper is devoted to the preliminary
results. In Section 3, we will present our main results and
proofs.
2. Preliminary Results
In this section we will introduce the notations in the
moving planes method and four results.
Let

be a real number,

is a bounded domain in
with a smooth boundary
n
R

,

11
max xx

  .
We choose a hyperplane perpendicular to the 1
x
direc-
tion in the process of moving a plane.
Consider the hyperplane


1
:n RxTx


 . For
sufficiently large positive number

. T

is disjoint
from . As we decrease


, at some moment, T

begins to intersect and from that moment T

will
cut off from an open cap 

1
xx


 . Let

  denote the reflection of


across the plane T

.
At the beginning of the process,


 will be contained
in , until one of the following events occurs: 
A1)


 becomes internally tangent to at some
point not on T
P

,
A2) T

reaches a position where it is orthogonal to
at some point .
 Q
We denote the hyperplane T

when it reaches either
one of these positions by T

and we call

 (an open
cap by T

) the maximal cap. Clearly its reflection



is contained in . 
Throughout this paper, the following notations will be
used: i
u
i
u
x

,
2
ij
u
ij
u
x
x



.
The following two lemmas are due to J. Serrin. We
will present the lemma, for detailed proof, see [8].
Lemma 2.1 Let be a domain with boundary
and be a hyperplane containing the normal to
2
C
T


at some point Q. Let denote the portion of



lying on some particular side of . T
Suppose that is of class in the closure of w2 C


and satisfies the elliptic inequality

,1
ij ij
ax w

0,
ij
x x

i
w
1
nn
i
i
bLw




ij
a


where the coefficients are uniformly bounded. We as-
sume that the matrix is uniformly definite
2
,1
ij k onst
 , 0kc
n
ij 
i j
ax  ,
and that

,1
, 0
n
ij ij
ij
aK dKconst  


 
,
where


1,, n

  is an arbitrary real vector,


,,
1n

 
d
is the unit normal to the plane T,
and is the distance from . Suppose also in
T0w


and 0w

at Q. Let
s
be any direction at
which enters
Q


nontangentially. Then
0
w
s

 or
2
20
w
s

 at Q,
unless 0w

.
Let

be a domain in ,
n
R

2
uC and 0u

.
Consider a linear differential operator of second order of
the form
 
,1 1
nn
ijijii
ij i
Luax ubxucxu



where is uniformly elliptic and the coefficients of
L are bounded. Here and in what follows, we use sum-
mation conventio
L
n.
The following lemma is a refinement of Hopf's bound-
ary point lemma (See [5]).
Lemma 2.2 Under the above assumptions, suppose
that 0Lu

and in , 0u 
1) If 0
x

 ,


00ux

and satisfies an interior
ball conditions at

0
x
, then we have

00
ux



where the vector


1,, n

  denotes the outer (unit
normal) to

 at 0
x
.
2) in
0u

.
Lemma 2.2 is well known for us in case


0cx

.
Here, we note the fact that it holds without the restriction
0c

.
Now we will present another key proposition which
describes the property near the boundary.
Proposition 2.3 Let

be a bounded domain of
class in . Let
2
Cn R
f
be of class . Assume that
1
C


uC
2

 is a solution of the problem


, in ufuu

 
,
0 in u,
0 on u

 .
If

is a real number which defines the maximal cap,
then for each point


01
xx

 , it holds either


10 0ux

or else


11 00ux
1
, that is, is a strictly
decreasing function of
u
x
near 0
x
.
Proof. Note that 10

 at 0
x
. By the hypothesis, it

Z. B. FANG ET AL. 342
is obvious that 0
u


 on

1
x


 , and so
10u on

1
x

 .
First, suppose


0, 0f0. Then we get


 


0,
,0,0
,0,0
ufuu
ufuu ff
ufuu f
 
 
 
0,0
Denote



12
12
,,,,,,,
n
nxxx
ppppuuu



,
f
ufuu
.
By the mean value theorem of multidimensional cal-
culus,
 
1
0
n
ii
f
fu fp




.
So for certain bounded functions , ,

i
bx

cx


 
1
0,
.
n
ii
i
ufuu f
ubxucx

 
 

0,0
Thus it follows that

00
ux



 by Lemma 2.2. Since
u is the opposite direction to the outward normal

on and
 10

 at 0
x
, we conclude

10 0ux

.
Now, suppose and ux and
by the hypotheses. We claim

0, 00

0
f

10 0
 
00,uxf



00, 0 iji j
ux f

  for each 1, (2.1) ij n
Since (2.1) is invariant under a rotation of coordinate
axes, we may consider a coordinate frame with origin at
0
x
, the 1
x
axis being directed along the outward nor-
mal to at 0

x
(at this moment,


1, 0,, 0  

). In
this frame we can represent the boundary of locally
by the equation

2
12
,,,
n
x
xx C  .
Since on , we have
0u

2
,,, 0
n
ux x 


 (2.2)
Differentiating (2.2) with respect to

2, ,
i
x
in,
we obtain
10 ii
uu

 (2.3)
Differentiating (2.3) with respect to

2, ,
j
x
jn,
we obtain

1111 1
0
iji jjjiij
uuu uu    (2.4)
Evaluating (2.4) at 0
x
, where 0
i

 and 10u

, we
find


00 ij
ux

for . 2,ij n
Thus




11 00, 0.ux f Since on 10u


1
x

  and


ux
10 0, the mapping




,,,
212
,,
nn
x
xu xx  has a maximum at 0
x
.
Hence


10 0 i
ux

for . So far we have
showed that (2.1) holds for . Returning to
the original coordinate axes, we can see that (2.1) holds
for each
2

1,ij n
in
1, 0

,, 0


f . Since and

0, 0010

, we
conclude


11 00ux.
Proposition 2.3 is proved.
Using the same argument, we obtain the following
corollary immediately.
Corollary 2.4 Let

be a bounded domain of class
in . Let
2
Cn R
f
be of class . Assume that
1
C


2
uC

 is a solution of the problem


in uf u

  ,
0 in u,
0 on u

 .
If

is a real number which defines the maximal cap,
then for each point

01
xx


 , it holds either


10 0ux

or else


11 00,ux that is, is a strictly
decreasing function of
u
1
x
near 0
x
.
Remark: In the Proposition 2.3, the hypothesis
on
1
C
f
can be replaced by either of the following condi-
tions:
1)




12

f
ufufu  for all , u
where 1
1
f
C

and 2
f
is nondecreasing,
2)
f
is locally Lipschitz continuous.
3. Main Results and Proofs
Now we will present our main results.
Theorem 3.1 Let
f
be a continuous function de-
fined on 1n
R

and satisfies the following conditions:
1)
f
is symmetric in 1
x
and nonincreasing in 1
x
for ,
1
x0
2) u
f
is continuous for . x
If there exists a positive solution in
u


2
uC


satisfying


,, in ufxuu

 ,
0 on u

 ,

on .
ucx





Then we have
1) There exists a real number

such that

and
are Steiner symmetric with respect to the hyperplane u

Z. B. FANG ET AL.343

1
x

.
2) For each , x

10x


implies that 1 x


,
and there exists a function
2
C:R


 such that
 
x 



x x   

,

1

1
x1
,2xx where



.
3) In the case when

is not zero,
f
is independ-
ent of the variable 1
x
in . 
Proof. For each real number

, where 1



,
we define functions and in v w

:


vx ux

 and . wvu
Then and satisfy, respectively
v w

,,, vfxvvx

,
, vuxT


,
0, .
c
vx T



and



,,,,,
0,
wfxvv fxuux
wx




a
 



(3.1)
By the mean value theorem, (3.1) can be rewritten into
 
,1 1
0
nn
ij iji i
ij i
axwbxwcx


 ,
where ij , i and are certain bounded functions.
Now we claim that for each real number
bc

1

  ,
 
vx ux in

, (3.2)
It follows from Proposition 2.3 that (3.2) holds for
each

sufficiently close to 1

.
Let


1
inf3.2 holds for each

   . We
will prove that

 . Assume

 . For any point
0\
x
T




0ux , we have . So we find
0
x



00
ux


w
.
Hence in 0

. Since in 0w

, applying
Lemma 2.2 to

 , we obtain
0w, x

 , and , .
10wxT 

So we find in
 
vx ux

 
 for sufficiently
small 0

.
We reach a contradiction with the choice of

.
Hence

  and this implies that
 
vx ux in

 for all

 ,
and




vx ux in

.
By Lemma 2.2, we see that either of the following
facts holds:
B1) at all interior points of 0w

, or else
B2) in 0w

.
Now, assume that 0w

in

, i.e., (B1) holds.
In the case (A1), by Lemma 2.2, we see

0
wP



,
where P

is the reflected value of across TP

. But
this is a contradiction, since
 
uv
P

P






. In the
case (A2), since we cannot apply Lemma 2.2 to the func-
tion , we make use of the substitution for . Let
, where k is a constant to be determined.
Then we obtain
w
ekx ww w
1


0w
 in


, at Q 0w

and
 
,1 1
0
nn
ij iji i
ij i
axwbxwcx






where ,
1i
2
ii
bb ka
2
11 1
cak bkc

 
k. For sufficiently large , we have
 
,1 1
0
nn
ij iji i
ij i
axwbxwcx





 in


.
By Lemma 2.1, it follows that
0
w
s


 or
2
20
w
s


 at Q.
At the point , since , we find
Q0w

1
ekx
ww
ss




 and 1
22
1
22
2e
kx
ww w
ks s
ss


 






.
Thus
0
w
s

 or
2
20
w
s

 at Q.
So far we have already showed that (B1) is impossible.
And obviously (B2) implies that is symmetric with
respect to ,
u
T

is Steiner symmetric with respect to
. Meanwhile, in
T0w

 for all

 , is
symmetrically decreasing in
u
1
x
.This completes the
proof of (1).
By the symmetry of a domain and the definition
of the maximal cap, for each ,

x

10x


im-
plies 1
x


. So 1 x


R
implies . So we can
see from the implicit function theorem that there exists a
function

x
1
 0
2
C:



 such that
  


11
,2
x
xx  xx 



ux

u
  
s
. Finally, the sym-
metry ofu implie

11
,x

,x

2x 

 in

. T hus
f
is symmetric with respect to the plane


1
x


in

. If 0


, by the hypothesis of
f
, we
can see that
f
is independent of the variable 1
x
in

. This completes the proof of the theorem.
Using Lemma 2.1, Lemma 2.2, Proposition 2.3, Cor-

Z. B. FANG ET AL. 344
ollary 2.4 and still the same argument, we can obtain the
following two corollaries.
Corollary 3.2 Let
f
be a continuous function de-
fined on 1n
R and satisfies the following conditions:
1)
f
ymmetric in 1 is s
x
and nonincreasing in 1
x
for 0x,
2)
1
i
u
f
is continuous for x

.
olutionIf t exists a positive s here

2
uC satisfy-
ing

, in ufxu ,
0 on u, 

on .
ucx





Then we have
a real number 1) There exists

such that

and
u are Steiner symmetric with respect to the hyperplane

1
x


.
2) For each x,

10x


implies that

1
x

, and xis function :there ets a 2 C R




such that
  

,2


11
x
xxxx  



x

, where

1



.
case when 3) In the

is not zero,
f
is independ-
ent of the variable 1
x
in.
Corollary 3.3 L et
f
ba fe unction class
fin
of 1 C de-
ed on R. If there exists a positive solution

2
uCsatisfying

in uf u ,
0 on u,

on .
ucx





Then we have
a real number 1) There exists

such that

and
u are Steiner symmetric with respect to the hyperplane


1
x

.
2) For each x,

10x


implies that


1
x

, and xis function :there ets a 2 C R




such that
  


11
,2
x
xxx   x

 
, where

1
x



 .
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