ABSTRACT

We show a result of symmetry for a big class of problems with condition of Neumann on the boundary in the case one dimensional. We use the method of reflection of Alexandrov and we show one application of this method and the maximum principle for elliptic operators in problems with conditions of Neumann. Some results of symmetry for elliptic problems with condition of Neumann on the boundary may be extended to elliptic operators more general than the Laplacian.

1. Introduction

The maximum principle is one of the most used tools in the study of some differential equations of elliptic type. It is a generalization of the following well-known theorem of the elemental calculus “If f is a function of class in such that the second derivative is positive on (a, b) then the maximum value of f attains at the ends of”. It is important to point out that the maximum principle gives information about the global behavior of a function over a domain from the information of qualitative character in the boundary and without explicit knowledge of the same function. The maximum principle allows us, for example, to obtain uniqueness of solution of certain problems with conditions of the Dirichlet and Neumann type. Also it allows to obtain a priori estimates for solutions. These reasons make interesting the study of the maximum principle on several forms and its generalizations and the Hopf lemma. For example a geometric version of the maximum principle allows us to compare locally surfaces that coincide at a point. On the other hand, the maximum principle and the Alexandrov reflection principle in [1] have been used to prove symmetries with respect to some point, some plane, symmetries of domain and to determine asymptotic-symmetric behavior of the solutions of some elliptic problems. (See Serrin [2], Gidas, Ni and Nirenberg [3], Gidas, Ni and Nirenberg [4], Caffarelli, Gidas and Spruck [5], Berestycki and Nirenberg [6]). The first person in use this technic was Serrin. Serrin proved that: “If is a positive solution of the problem

which is zero on the boundary and its outer normal derivative on the boundary is constant, then is a ball and is radially symmetric with respect to the center of”. Using the ideas of Serrin and a version of the maximum principle for functions that do not change of sign, Gidas Ni and Nirenberg proved that: “If is a ball, and is a positive solution of the problem,

which is zero on the boundary, then u is radially symmetric with respect to the center of the ball”. Using the method of reflection and a version of maximum principle for thin domains Berestycki and Nirenberg made a generalization of this statement. Our proof shows that the technic used by Berestycki and Nirenberg for the study of symmetries of solutions of the elliptic problem with Dirichlet condition, can be applied in elliptic problems with Neumann conditions with nonlinear term.

2. Maximum Principle and Hopf Lemma

Our result is based on the well known maximum principle and on the Hopf lemma for the differential operator of the form (see [7-9])

(1)

where is in We suppose that the coefficients and are bounded on and for all

Theorem 2.1. (Maximum principle)

Let be such that Then cannot attain its maximum value in

Lemma 2.2. (Hopf)

Suppose satisfies

Let be such that 

• is continuous at

• for all

• existe.

Then

3. Main Result

Theorem 3.3. Let be a solution of

where are bounded functions and symmetric with respect to the origin such that

and for all is such that is strictly increasing in for all and is symmetric to for all and is a bounded function and odd. Then is symmetric with respect to the origin.

Proof: Define the reflected function of in by

Hence, Then v satisfies

Define

Then satisfies

Since is continuous in, there are such that

Suppose that or then if

since Further

Therefore

Since is strictly increasing in

Then

Therefore

If using a similar argue we demonstrate that and we obtain the same conclusion. Suppose that then since w(0) = 0 Further Therefore

Since is strictly increasing in

Then

Therefore

We conclude

So is symmetric with respect to the origin.

We will prove that do not belong to Suppose now that and

for all then

and If and

then and where are such that is the first zero of w and is the last. Since is strictly increasing in, then

and

Applying maximum principle and Hopf lemma,

since is not constant. Which contradicts the fact that

Hence this case is impossible. It happens equally to and In conclusion we have that on and therefore is symmetric with respect to

4. Example

Taking

in Theorem 3.3, we have the following system

following the steps of the demonstration, it follows that u is symmetric with respect to the origin.

5. Acknowledgements

The authors express their deep gratitude to CONACYT México, Programa de Mejoramiento del Profesorado (PROMEP)-México and Universidad de Cartagena for financial support.

REFERENCES