Symmetry of solutions to parabolic Monge-Ampère equations
© Dai; licensee Springer 2013
Received: 3 April 2013
Accepted: 5 August 2013
Published: 20 August 2013
In this paper, we study the parabolic Monge-Ampère 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 and an example in a cylinder to illustrate the results.
The Monge-Ampère 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 Monge-Ampère 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 Monge-Ampère mapping problem . More applications of the Monge-Ampère equation and the optimal transportation can be found in [3, 4]. In the meantime, the Monge-Ampère equation turned out to be the prototype for a class of questions arising in differential geometry.
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.
The basic technique they applied is the method of moving planes first introduced by Alexandrov  and then developed by Serrin . Later the symmetry results of elliptic equations have been generalized and extended by many authors. Especially, Li  considered fully nonlinear elliptic equations on smooth domains, and Berestycki and Nirenberg  found a way to deal with general equations with nonsmooth domains using the maximum principles on domains with small measure. Recently, Zhang and Wang  investigated the symmetry of the elliptic Monge-Ampère equation and they got the following results.
has the above symmetry and monotonicity properties (1.4) and (1.5). Extensions in various directions including degenerate problems  or elliptic systems of equations  were studied by many authors.
The result of is as follows.
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
We use the standard notation to denote the class of functions u such that the derivatives are continuous in Q for .
then in Q.
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.
Then in Q.
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. □
then we have the following narrow region principle.
Corollary 2.3 (Narrow region principle)
Suppose that satisfies (2.2) and (2.3). Let the width l of Ω be sufficiently small. If on , , then we have in Q. If Ω is unbounded, and , then the conclusion is also true.
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 is called parabolically convex if it is continuous, convex in x and decreasing in t.
is bounded in .
where , ().
Theorem 3.1 Let Ω be a strictly convex domain in and symmetric with respect to the plane , . Assume that conditions (A) and (B) hold and is any parabolically convex solution of (1.1), (1.2) and (1.3). Then , where , and when , .
From Corollary 2.3, when the width of is sufficiently small, , .
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.
This contradicts with the definition of Λ, and so .
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 3.2 If Ω is a ball, , then any parabolically convex solution of (1.1), (1.2) and (1.3) is radially symmetric about the origin.
then u is a solution of (3.15) but not radially symmetric.
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). □
The research was supported by NNSFC (11201343), Shandong Province Young and Middle-Aged Scientists Research Awards Fund (BS2011SF025), Shandong Province Science and Technology Development Project (2011YD16002).
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