﻿ Some Symmetry Results for the A-Laplacian Equation via the Moving Planes Method

Vol.2 No.6(2012), Article ID:24363,4 pages DOI:10.4236/apm.2012.26053

Some Symmetry Results for the A-Laplacian Equation via the Moving Planes Method*

Zhongbo Fang, Anna Wang

School of Mathematical Sciences, Ocean University of China, Qingdao, China

Email: fangzb7777@hotmail.com

Received July 3, 2012; revised September 1, 2012; accepted September 8, 2012

Keywords: Symmetry; A-Laplacian; Moving Planes Method; Overdetermined Boundary Value Problem

ABSTRACT

In this paper, we are concerned with a positive solution of the non-homogeneous A-Laplacian equation in an open bounded connected domain. We use moving planes method to prove that the domain is a ball and the solution is radially symmetric.

1. Introduction

In this paper, we are going to study the symmetry results for the overdetermined problem

(1.1)

(1.2)

(1.3)

Here is a bounded connected open subset of with boundary and is a point in. The function satisfies the regularity requirement

(1.4)

and the (possibly degenerate) elliptic condition

(1.5)

is a continuously differentiable function. is a constant and denotes the inner normal to.

J. Serrin proved the radial symmetry for positive solutions of the equation in with the same overdetermined boundary conditions as the above problem, see [1]. N. Garofalo and J. Lewis extended Serrin’s result to a larger class of elliptic equations possibly degenerate, including the following p-Laplacian equation

with the same boundary conditionssee [2]. For the overdetermined elliptic boundary value problem in with the same overdetermined boundary conditions as above, I. Fragala, I. F. Gazzaola and B. Kawohl used the geometric approach which relies on a maximum principle for a suitable Pfunction, combined with some geometric arguments involving the mean curvature of to prove that if the above problem admits a solution in a suitable weak sense, then is a ball, see [3]. A. Farina and B. Kawohl obtained the same result under removing the strong ellipticity assumption in [4] and a growth assumption in [2] on the diffusion coefficient A, as well as a starshapedness assumption on in [3], see [5]. A. Firenze considered the positive solution of problem (1.1)-(1.3) when it is a p-Laplacian equation in an open bounded connected subset of with boundary, see [6]. All of the above motivated us to extend the symmetry result to the non-homogeneous A-Laplacian equation.

Our main result is that for the problem (1.1)-(1.3), if u has only one critical point in, then is a ball and u is radially symmetric.

Section 2 of this paper is devoted to the main result and a more general version of this theorem. In Section 3, we will present the proof of the main theorem.

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2. Preliminaries and Statement of Results

In this section we give some lemma that we shall use and present our main result.

Lemma 2.1. (The boundary lemma at corner) (Lemma 2 in [1]) Let be a domain with C2 boundary and be a hyperplane containing the normal to at some point. Let denote the portion of lying on some particular side of.

Suppose that is of class in the closure of and satisfies the elliptic inequality

,

where the coefficients are uniformly bounded. We assume that the matrix is uniformly definite

, and that

, where is an arbitrary real vector, is the unit normal to the plane, and is the distance from. Suppose also in and at. Let be any direction at which enters nontangentially. Then

or atunless.

Our main results are as follows:

Theorem 2.2. Let be a bounded connected open subset of with boundary and let be a point in

. Let, , be a strictly positive solution of the following overdetermined boundary value problem

(2.1)

(2.2)

(2.3)

Here is a continuously differentiable function, and

. (2.4)

c is a constant and denotes the inner normal to. Assume

(2.5)

then is a ball and is radially symmetric.

The following remark is a general version of the theorem. It can be viewed as an extension result of p-Laplacian too. As the proof is similar to Theorem 2.2, we omit it.

Remark 2.3. Let be as in Theorem 2.2 and D be a subset of. Let be a strictly positive solution of Equation (2.1) in and verify the boundary conditions; Assume that is the critical set of, then if denotes the convex hull of1) the normal line to at an arbitrary point of intersects;

2) if is a support plane to through and is a ray from A orthogonal to which lies in the half-space determined by not containing, then intersects exactly in one point.

In what follows we assume that the origin of the coordinates system is an interior point of, and we denote with the closure of the ball centered in with radius.

Theorem 2.4. Assume that the hypotheses of Theorem 2.2 hold and furthermore assume that

for some positive. Then 1) is starshaped with respect to;

2) if

;

;

then

.

3. Proof of Theorem 2.1

The technique we are going to use is the moving planes method. For the detailed description about moving planes method, see [1].

Proof. Step 1: To prove is a ball.

If we can demonstrate that for any point Q on, P lies on the normal line to at Q, then is a ball with centre P. To do this, we argue by contradiction.

Assume that there exists a point such that the normal line to at Q does not contain P. We choose a coordinate system in such that , , and the xn axis coincides with r.

When we use the moving planes method, we choose a family of hyperplanes normal to the axis. Define hyperplan for any positive; Let be the infimum of such that; Define for and we denote by the reflection in. Since is, for some close to, v, we have

(3.1)

As decreases, condition (3.1) holds until one of the following facts happens:

1) is internally tangent to at some point of;

2) intersects at some point of.

Let be the greatest value of, , such that either condition a) or b) is true. Since is orthogonal to at, we have and then for any in. This is the crucial point of our proof. We have found a direction such that as the moving plane moves from to the critical position, it never intersects, so that the moving planes method may be applied.

Let be the reflected point of x in. We defined

for, ,

From Equation (2.1) we have for,

(3.2)

By the definition of v, we obtain

(3.3)

Differencing Equations (3.2) and (3.3) yields

(3.4)

Meanwhile, (3.4) can also be rewritten into

(3.5)

Denote, ,

.

Let

By the mean value theorem, it follows from (3.5) that

(3.6)

where, and c are certain functions depending on u and f. Here the matrix is uniformly positive definite, since both expressions and have this property (recall that Equation (2.1) is elliptic). So (3.6) is uniformly elliptic with bounded coefficients far from, i.e. in where is a ball centered in with radius, for any positive.

From the boundary condition (2.3) on the normal derivative of, it follows that

in (3.7)

for some sufficiently close to. Let

. We prove.

Assume, by continuity, in. On the other hand, since is not symmetric with respect to, in. By the strong version of the maximum principle, we obtain in

. Next we observe that can not be a critical point for w since while

. So as is arbitrarily small, it is

in. Since, we may apply the Hopf lemma to at each point of, we get

on           (3.8)

The plane is not normal to at any point, then from inequality (3.8) and the boundary condition (2.3) on the normal derivative of, we get

(3.9)

By the definition of, there exists a sequence such that and

(3.10)

Let be a limit point for xn in the closure ofby continuity, thus. But from inequality (3.10) and the mean value theorem we get and this contradicts condition (3.9).

So is proved.

Now we will prove that u must be symmetric with respect to. Assume in, so as we did for, we infer in.

Assume next that condition a) holds, then is internally tangent to at some point, where. Since P is an interior point of, , so that we can apply the Hopf lemma to w at M and we obtain

;

where is the inner normal to at M. For

we get the contradiction. Hence condition 2) must be true, i.e. is orthogonal to at some point B. From the boundary condition (2.3) and the definition of w it follows that all the first and second derivatives of w vanish at B. On the other hand, as, Equation (3.6) is uniformly elliptic with bounded coefficents in a neighborhood of B, so that the boundary lemma at corner in [1] lemma 2, may be applied to w. Let s be a direction which enters nontangentially at B, then by the Serrin’s lemma

or

Then we have again a contradiction with the derivatives of w at B, so in. But this last inequality can not be true since otherwise w would be a function symmetric in whose only critical point is not on.

This completes the proof of Theorem 2.1.

REFERENCES

1. J. Serrin, “A Symmetry Problem in Potential Theory,” Archive for Rational Mechanics and Analysis, Vol. 43, No. 4, 1971, pp. 304-318. doi:10.1007/BF00250468
2. N. Garofalo and J. Lewis, “A Symmetry Result Related to Some Overdetermined Boundary Value Problems,” American Journal of Mathematics, Vol. 111, No. 1, 1989, pp. 9-33. doi:10.2307/2374477
3. I. Fragala, I. F. Gazzaola and B. Kawohl, “Overdetemined Boundary Value Problems with Possibly Degenerate Ellipticity: A Geometry Approach,” Mathematische Zeitschrift, Vol. 254, No. 1, 2006, pp. 117-132. doi:10.1007/s00209-006-0937-7
4. G. A. Philippin, “Application of the Maximum Principle to a Variety of Problems Involving Elliptic Differential Equations,” In: P. W. Schaefer, Ed., Maximum Principles and Eigenvalue Problems in Partial Differential Equations, Pitman Research Notes in Mathematics Series, Longman SciTech., Harlow, 1988, pp. 34-48.
5. A. Farina and B. Kawohl, “Remarks on an Overdetermined Boundary Value Problem,” Calculus of Variations and Partial Differential Equations, Vol. 31, No. 3, 2008, pp. 351-357.
6. A. Firenze, “A Symmetry Result for the p-Laplacian Equation via the Moving Planes Method,” Applocable. Analysis, Vol. 55, No. 3-4, 1994, pp. 207-213.

NOTES

*This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018).