Some Symmetry Results for the A-Laplacian Equation via the Moving Planes Method ()
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
, ![](https://www.scirp.org/html/1-5300169\63105374-f331-4563-a6c4-bb45c35696c2.jpg)
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
at
unless
.
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 of
1) 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
![](https://www.scirp.org/html/1-5300169\f6aa6da6-aa86-4ac9-89ff-34562bcd9339.jpg)
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
,
,
![](https://www.scirp.org/html/1-5300169\2f86426d-67f4-45fd-9e95-e1a5ec59f61e.jpg)
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
![](https://www.scirp.org/html/1-5300169\9cfa898f-18ad-4ce7-95a1-2a63fdce3fed.jpg)
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 ![](https://www.scirp.org/html/1-5300169\0257afbf-4066-4564-a140-79b569e2fa47.jpg)
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 of
by 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
![](https://www.scirp.org/html/1-5300169\f135d180-225f-4598-b057-4a63bff33122.jpg)
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 ![](https://www.scirp.org/html/1-5300169\431431eb-4c3d-4ce8-b497-cd6bc2cd125c.jpg)
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.
NOTES