Abstract
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 was first introduced by Krylov [12] together with the other parabolic versions of elliptic MongeAmpère equations; see [8] for a complete description and related results. It is also relevant in the study of deformation of surfaces by GaussKronecker curvature [13,14] and in a maximum principle for parabolic equations [15]. Tso [15] pointed out that the parabolic equation is the most appropriate parabolic version of the elliptic MongeAmpère equation in the proof of AleksandrovBakelman maximum principle of secondorder parabolic equations. In this paper, we study the symmetry of solutions to the parabolic MongeAmpère equation
where is the Hessian matrix of u in x, , Ω is a bounded and convex open subset in , denotes the side of Q, denotes the bottom of Q, and denotes the parabolic boundary of Q, f and are given functions.
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 or Ω is a smooth bounded domain in , convex in and symmetric with respect to the hyperplane , then any positive solution of the Dirichlet problem
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 and they got the following results.
Let Ω be a bounded convex domain in with smooth boundary and symmetric with respect to the hyperplane , then each solution of the Dirichlet problem
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 , , Gidas et al.[16] studied parabolic equations and , and they proved that parabolic equations possessed the same symmetry as the above elliptic equations. When , Hess and Poláčik [25] first studied the asymptotic symmetry results for classical, bounded, positive solutions of the problem
The symmetry of general positive solutions of parabolic equations was investigated in [24,26,27] and the references therein. A typical theorem of is as follows.
Let Ω be convex and symmetric in . If u is a bounded positive solution of (1.6) and (1.7) with satisfying
then u has the symmetry and monotonicity properties for each :
Assume that u is a bounded positive solution of (1.6) and (1.7) with such that for some sequence ,
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 , let , , be continuous functions in . Suppose that , is bounded and there exist positive constants and such that
Here and in the sequel, we always denote
We use the standard notation to denote the class of functions u such that the derivatives are continuous in Q for .
Theorem 2.1Letbe a bounded continuous function on, and let the positive functionsatisfy
If
Proof We argue by contradiction. Suppose there exists such that . Let
Then . Set , then and . Since attains its minimum at , we have , . In addition, we have . A direct calculation gives
Taking into account , we have at ,
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, . Besides the conditions of Theorem 2.1, we assume
Proof Still consider in the proof of Theorem 2.1. Condition (2.5) shows that the minimum of cannot be achieved at infinity. The rest of the proof is the same as the proof of Theorem 2.1. □
If Ω is a narrow region with width l,
then we have the following narrow region principle.
Corollary 2.3 (Narrow region principle)
Suppose thatsatisfies (2.2) and (2.3). Let the widthlof Ω be sufficiently small. If on, , then we havein Q. If Ω is unbounded, and, then the conclusion is also true.
Then φ is positive and
Choose . In virtue of the boundedness of , when l is sufficiently small, we have , and thus
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 is called parabolically convex if it is continuous, convex in x and decreasing in t.
Suppose that the following conditions hold.
Theorem 3.1Let Ω be a strictly convex domain inand symmetric with respect to the plane, . Assume that conditions (A) and (B) hold andis any parabolically convex solution of (1.1), (1.2) and (1.3). Then, where, and when, .
Then
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
is the inverse matrix of . Then , is bounded and by the a priori estimate [9] we know there exist positive constants and such that
Let
then from (3.5),
Clearly,
Because the image of about the plane lies in Ω, according to the maximum principle of parabolic MongeAmpère equations,
Thus
On the other hand, from (3.1),
From Corollary 2.3, when the width of is sufficiently small, , .
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 , then the image of under the reflection about lies inside Ω. According to the strong maximum principle of parabolic MongeAmpère equations, for , . Therefore, for , we have . On the other hand, by the definition of Λ, we have for , . So, from the strong maximum principle [28] of linear parabolic equations and (3.6), we have for ,
Let be the maximum width of narrow regions so that we can apply the narrow region principle. Choose a small positive constant δ such that , . We consider the function on the narrow region
Now we prove the boundary condition
Similar to boundary conditions (3.7), (3.8) and (3.9), boundary condition (3.12) is satisfied for , and for . In order to prove (3.12) is satisfied for , we apply the continuity argument. By (3.10) and the fact that is inside , there exists a positive constant such that
Because is continuous in λ, then for small δ, we still have
Therefore boundary condition (3.12) holds for small δ. From Corollary 2.3, we have
Combining (3.10) and the fact that is continuous for λ, we know that for when δ is small. Then from (3.13), we know that
This contradicts with the definition of Λ, and so .
As a result, for , which means that as ,
Since Ω is symmetric about the plane , then for , also satisfies (1.1). Thus we can move the plane from the right towards the left and get the reverse inequality. Therefore
Equation (3.14) means that u is symmetric about the plane . Theorem 3.1 is proved. □
If we put the axis in any direction, from Theorem 3.1, we have the following.
Corollary 3.2If Ω is a ball, , then any parabolically convex solutionof (1.1), (1.2) and (1.3) is radially symmetric about the origin.
Remark 3.1 Solutions of (1.1) in may not be radially symmetric. For example,
has a nonradially symmetric solution. In fact, we know that () satisfies in , and , . Define
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 be the unit ball in , and let radially symmetric function , satisfy
Example 3.1 Let satisfy (3.16) and (3.17). Then any solution of
is of the form
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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