Abstract
In this paper, we study the parabolic MongeAmpère equation
Using the method of moving planes, we show that any parabolically convex solution
is symmetric with respect to some hyperplane. We also give a counterexample in
MSC: 35K96, 35B06.
Keywords:
parabolic MongeAmpère equations; symmetry; method of moving planes1 Introduction
The MongeAmpère equation has been of much importance in geometry, optics, stochastic theory, mass transfer problem, mathematical economics and mathematical finance theory. In optics, the reflector antenna system satisfies a partial differential equation of MongeAmpère type. In [1,2], Wang showed that the reflector antenna design problem was equivalent to an optimal transfer problem. An optimal transportation problem can be interpreted as providing a weak or generalized solution to the MongeAmpère mapping problem [3]. More applications of the MongeAmpère equation and the optimal transportation can be found in [3,4]. In the meantime, the MongeAmpère equation turned out to be the prototype for a class of questions arising in differential geometry.
For the study of elliptic MongeAmpère equations, we can refer to the classical papers
[57] and the study of parabolic MongeAmpère equations; see the references [811]etc. The parabolic MongeAmpère equation
where
There is vast literature on symmetry and monotonicity of positive solutions of elliptic
equations. In 1979, Gidas et al.[16] first studied the symmetry of elliptic equations, and they proved that if
satisfies the following symmetry and monotonicity properties:
The basic technique they applied is the method of moving planes first introduced
by Alexandrov [17] and then developed by Serrin [18]. Later the symmetry results of elliptic equations have been generalized and extended
by many authors. Especially, Li [19] considered fully nonlinear elliptic equations on smooth domains, and Berestycki and
Nirenberg [20] found a way to deal with general equations with nonsmooth domains using the maximum
principles on domains with small measure. Recently, Zhang and Wang [21] investigated the symmetry of the elliptic MongeAmpère equation
Let Ω be a bounded convex domain in
has the above symmetry and monotonicity properties (1.4) and (1.5). Extensions in various directions including degenerate problems [22] or elliptic systems of equations [23] were studied by many authors.
For the symmetry results of parabolic equations on bounded and unbounded domains,
the reader can be referred to [16,24,25] and the references therein. In particular, when
The symmetry of general positive solutions of parabolic equations was investigated
in [24,26,27] and the references therein. A typical theorem of
Let Ω be convex and symmetric in
then u has the symmetry and monotonicity properties for each
The result of
Assume that u is a bounded positive solution of (1.6) and (1.7) with
Then u is asymptotically symmetric in the sense that
In this paper, using the method of moving planes, we obtain the same symmetry of solutions to problem (1.1), (1.2) and (1.3) as elliptic equations.
2 Maximum principles
In this section, we prove some maximum principles. Let Ω be a bounded domain in
Here and in the sequel, we always denote
We use the standard notation
Theorem 2.1Let
Suppose that
If
then
Proof We argue by contradiction. Suppose there exists
Then
Taking into account
This is a contradiction and thus completes the proof of Theorem 2.1. □
Theorem 2.1 is also valid in unbounded domains if u is nonnegative at infinity. Thus we have the following corollary.
Corollary 2.2Suppose that Ω is unbounded,
Then
Proof Still consider
If Ω is a narrow region with width l,
then we have the following narrow region principle.
Corollary 2.3 (Narrow region principle)
Suppose that
Proof Let
Then φ is positive and
Choose
From Theorem 2.1, we have
3 Main results
In this section, we prove that the solutions of (1.1), (1.2) and (1.3) are symmetric by the method of moving planes.
Definition 3.1 A function
Suppose that the following conditions hold.
(A)
(B)
where
Theorem 3.1Let Ω be a strictly convex domain in
Proof Let in
Then
where
We rewrite (3.2) in the form
On the other hand, from (1.1), we have
According to (3.3) and (3.4), we have
Therefore
As a result, we have
where
Let
then from (3.5),
Clearly,
Because the image of
Thus
On the other hand, from (3.1),
From Corollary 2.3, when the width of
Now we start to move the plane to its right limit. Define
We claim that
Otherwise, we will show that the plane can be further moved to the right by a small distance, and this would contradict with the definition of Λ.
In fact, if
Let
Then
Now we prove the boundary condition
Similar to boundary conditions (3.7), (3.8) and (3.9), boundary condition (3.12)
is satisfied for
Because
Therefore boundary condition (3.12) holds for small δ. From Corollary 2.3, we have
Combining (3.10) and the fact that
This contradicts with the definition of Λ, and so
As a result,
Since Ω is symmetric about the plane
Equation (3.14) means that u is symmetric about the plane
If we put the
Corollary 3.2If Ω is a ball,
Remark 3.1 Solutions of (1.1) in
has a nonradially symmetric solution. In fact, we know that
then u is a solution of (3.15) but not radially symmetric.
We conclude this paper with a brief examination of Theorem 3.1. Let
Example 3.1 Let
is of the form
where
Proof According to Corollary 3.2, the solution is symmetric. Let
Then
Therefore (3.18) is
We seek the solution of the form
Then
That is,
Therefore
By (3.20), we know that
From (3.24) and (3.25), we have
As a result,
From the maximum principle, we know that the solution of (3.18)(3.20) is unique. Thus any solution of (3.18), (3.19) and (3.20) is of the form of (3.21). □
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The research was supported by NNSFC (11201343), Shandong Province Young and MiddleAged Scientists Research Awards Fund (BS2011SF025), Shandong Province Science and Technology Development Project (2011YD16002).
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