Some Steiner Symmetry Results in Overdetermined Boundary Value Problem *

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.


Introduction
We will present some symmetry results in overdetermined boundary value problem , , , n x x x x    and  is the outward unit normal to .Here is a bounded domain of class in , f is a function of class and satisfies certain conditions, is a differentiable function.Obviously the constant normal derivative of on is a special case of this problem.
1 C c 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 described by the following overdetermined problem where is a bounded domain of class in such that there exists a function satisfying the above problem, and  is the unit normal to .Using the moving planes method, J. Serrin proved that if the above problem has a solution in , 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, 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, 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][6][7][8][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 boundary problem containing gradient.
We shall prove that Section 2 of this paper is devoted to the preliminary results.In Section 3, we will present our main results and proofs.

Preliminary Results
In this section we will introduce the notations in the moving planes method and four results.Let The following two lemmas are due to J. Serrin.We will present the lemma, for detailed proof, see [8].
where the coefficients are uniformly bounded.We assume that the matrix is uniformly definite and that

 
, 1 , 0 is the unit normal to the plane T , and is the distance from .Suppose also in T 0 w    and 0 w  at Q .Let s be any direction at which enters Consider a linear differential operator of second order of the form where is uniformly elliptic and the coefficients of L are bounded.Here and in what follows, we use summation conventio The following lemma is a refinement of Hopf's boundary point lemma (See [5]).where the vector Lemma 2.2 is well known for us in case   0 c x  .Here, we note the fact that it holds without the restriction 0 c  .Now we will present another key proposition which describes the property near the boundary.
If  is a real number which defines the maximal cap, then for each point , and so , , , , , , , By the mean value theorem of multidimensional calculus, So for certain bounded functions , , .
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 normal to at 0  x (at this moment, ).In this frame we can represent the boundary of locally by the equation Evaluating (2.4) at 0 x , where 0 i   and 1 0 u  , we find If  is a real number which defines the maximal cap, then for each point x .Remark: In the Proposition 2.3, the hypothesis on 1 C f can be replaced by either of the following conditions:

Main Results and Proofs
Now we will present our main results.
Theorem 3.1 Let f be a continuous function defined on 1 n R  and satisfies the following conditions: 1) f is symmetric in 1 x and nonincreasing in 1 x for , Then we have 1) There exists a real number  such that  and are Steiner symmetric with respect to the hyperplane u and there exists a function 3) In the case when  is not zero, f is independent of the variable 1 x in . Proof.For each real number  , where 1      , we define functions and in v w By the mean value theorem, (3.1) can be rewritten into , where ij , i and are certain bounded functions.Now we claim that for each real number It follows from Proposition 2.3 that (3.2) holds for each  sufficiently close to 1  . Let for sufficiently small 0   .We reach a contradiction with the choice of  .Hence    and this implies that and By Lemma 2.
. For sufficiently large , we have By Lemma 2. x .This completes the proof of (1).
By the symmetry of a domain and the definition of the maximal cap, for each , . So we can see from the implicit function theorem that there exists a function Then we have a real number 1) There exists  such that  and u are Steiner symmetric with respect to the hyperplane

Lemma 2 . 2
Under the above assumptions, suppose that

2 )
For each x   , is not zero, f is independent of the variable 1x in  .Corollary 3.3 L et f b a such that  and u are Steiner symmetric with respect to the hyperplane by the hypothesis of f , we can see that f is independent of the variable 1x in  .This completes the proof of the theorem.