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Symmetry of solutions to parabolic Monge-Ampère equations

Boundary Value Problems20132013:185

https://doi.org/10.1186/1687-2770-2013-185

Received: 3 April 2013

Accepted: 5 August 2013

Published: 20 August 2013

Abstract

In this paper, we study the parabolic Monge-Ampère equation

u t det ( D 2 u ) = f ( t , u ) in  Ω × ( 0 , T ] .

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 R n × ( 0 , T ] and an example in a cylinder to illustrate the results.

MSC:35K96, 35B06.

Keywords

  • parabolic Monge-Ampère equations
  • symmetry
  • method of moving planes

1 Introduction

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 [3]. 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.

For the study of elliptic Monge-Ampère equations, we can refer to the classical papers [57] and the study of parabolic Monge-Ampère equations; see the references [811]etc. The parabolic Monge-Ampère equation u t det ( D 2 u ) = f was first introduced by Krylov [12] together with the other parabolic versions of elliptic Monge-Ampère equations; see [8] for a complete description and related results. It is also relevant in the study of deformation of surfaces by Gauss-Kronecker curvature [13, 14] and in a maximum principle for parabolic equations [15]. Tso [15] pointed out that the parabolic equation u t det ( D 2 u ) = f is the most appropriate parabolic version of the elliptic Monge-Ampère equation det ( D 2 u ) = f in the proof of Aleksandrov-Bakelman maximum principle of second-order parabolic equations. In this paper, we study the symmetry of solutions to the parabolic Monge-Ampère equation
u t det ( D 2 u ) = f ( t , u ) , ( x , t ) Q ,
(1.1)
u = 0 , ( x , t ) S Q ,
(1.2)
u = u 0 ( x ) , ( x , t ) B Q ,
(1.3)

where D 2 u is the Hessian matrix of u in x, Q = Ω × ( 0 , T ] , Ω is a bounded and convex open subset in R n , S Q = Ω × ( 0 , T ) denotes the side of Q, B Q = Ω ¯ × { 0 } denotes the bottom of Q, and p Q = S Q B Q denotes the parabolic boundary of Q, f and u 0 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 Ω = R n or Ω is a smooth bounded domain in R n , convex in x 1 and symmetric with respect to the hyperplane { x R n : x 1 = 0 } , then any positive solution of the Dirichlet problem
Δ u + f ( u ) = 0 , x Ω , u = 0 , x Ω
satisfies the following symmetry and monotonicity properties:
u ( x 1 , x 2 , , x n ) = u ( x 1 , x 2 , , x n ) ,
(1.4)
u x 1 ( x 1 , x 2 , , x n ) < 0 ( x 1 > 0 ) .
(1.5)

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 Monge-Ampère equation det ( D 2 u ) = e u and they got the following results.

Let Ω be a bounded convex domain in R n with smooth boundary and symmetric with respect to the hyperplane { x R n : x 1 = 0 } , then each solution of the Dirichlet problem
det ( D 2 u ) = e u , x Ω , u = 0 , x Ω

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 Q = Ω × J , J = ( 0 , T ] , Gidas et al. [16] studied parabolic equations u t + Δ u + f ( t , r , u ) = 0 and u t + F ( t , x , u , D u , D 2 u ) = 0 , and they proved that parabolic equations possessed the same symmetry as the above elliptic equations. When J = ( 0 , ) , Hess and Poláčik [25] first studied the asymptotic symmetry results for classical, bounded, positive solutions of the problem
u t Δ u = f ( t , u ) , ( x , t ) Ω × J ,
(1.6)
u = 0 , ( x , t ) Ω × J .
(1.7)

The symmetry of general positive solutions of parabolic equations was investigated in [24, 26, 27] and the references therein. A typical theorem of J = R is as follows.

Let Ω be convex and symmetric in x 1 . If u is a bounded positive solution of (1.6) and (1.7) with J = R satisfying
inf t R u ( x , t ) > 0 ( x Ω , t J ) ,
then u has the symmetry and monotonicity properties for each t R :
u ( x 1 , x , t ) = u ( x 1 , x , t ) ( x = ( x 1 , x ) Ω , t R ) , u x 1 ( x , t ) < 0 ( x Ω , x 1 > 0 , t R ) .

The result of J = ( 0 , ) is as follows.

Assume that u is a bounded positive solution of (1.6) and (1.7) with J = ( 0 , ) such that for some sequence t n ,
lim inf n u ( x , t n ) > 0 ( x Ω ) .
Then u is asymptotically symmetric in the sense that
lim t ( u ( x 1 , x , t ) u ( x 1 , x , t ) ) = 0 ( x Ω ) , lim sup t u x 1 ( x , t ) 0 ( x Ω , x 1 > 0 ) .

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 R n , let a i j ( x , t ) , b ( x , t ) , c ( x , t ) be continuous functions in Q ¯ , Q = Ω × ( 0 , T ] . Suppose that b ( x , t ) < 0 , c ( x , t ) is bounded and there exist positive constants λ 0 and Λ 0 such that
λ 0 | ξ | 2 a i j ( x , t ) ξ i ξ j Λ 0 | ξ | 2 , ξ R n .
Here and in the sequel, we always denote
D i = x i , D i j = 2 x i x j .

We use the standard notation C 2 k , k ( Q ) to denote the class of functions u such that the derivatives D x i D t j u are continuous in Q for i + 2 j 2 k .

Theorem 2.1 Let λ ( x , t ) be a bounded continuous function on Q ¯ , and let the positive function φ C 2 , 1 ( Q ¯ ) satisfy
b ( x , t ) φ t + a i j ( x , t ) D i j φ λ ( x , t ) φ 0 .
(2.1)
Suppose that u C 2 , 1 ( Q ) C 0 ( Q ¯ ) satisfies
b ( x , t ) u t + a i j ( x , t ) D i j u c ( x , t ) u 0 , ( x , t ) Q ,
(2.2)
u 0 , ( x , t ) p Q .
(2.3)
If
c ( x , t ) > λ ( x , t ) , ( x , t ) Q ,
(2.4)

then u 0 in Q.

Proof We argue by contradiction. Suppose there exists ( x ¯ , t ¯ ) Q such that u ( x ¯ , t ¯ ) < 0 . Let
v ( x , t ) = u ( x , t ) φ ( x , t ) , ( x , t ) Q .
Then v ( x ¯ , t ¯ ) < 0 . Set v ( x 0 , t 0 ) = min Q ¯ v ( x , t ) , then x 0 Ω and v ( x 0 , t 0 ) < 0 . Since v ( , t 0 ) attains its minimum at x 0 , we have D v ( x 0 , t 0 ) = 0 , D 2 v ( x 0 , t 0 ) 0 . In addition, we have v t ( x 0 , t 0 ) 0 . A direct calculation gives
v t = u t φ u φ t φ 2 , D i j v = 1 φ D i j u u φ 2 D i j φ 1 φ D i v D j φ 1 φ D j v D i φ .
Taking into account u ( x 0 , t 0 ) < 0 , we have at ( x 0 , t 0 ) ,
0 φ a i j D i j v = a i j D i j u a i j D i j φ φ u a i j D i j u + u φ ( b φ t λ φ ) a i j D i j u + b φ u t φ λ u = a i j D i j u + b u t λ u < a i j D i j u + b u t c u 0 .

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 2.2 Suppose that Ω is unbounded, Q = Ω × ( 0 , T ] . Besides the conditions of Theorem 2.1, we assume
lim inf | x | u ( x , t ) 0 .
(2.5)

Then u 0 in Q.

Proof Still consider v ( x , t ) in the proof of Theorem 2.1. Condition (2.5) shows that the minimum of v ( x , t ) 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,
Ω = { x R n | 0 < x 1 < l } ,

then we have the following narrow region principle.

Corollary 2.3 (Narrow region principle)

Suppose that u C 2 , 1 ( Q ) C 0 ( Q ¯ ) satisfies (2.2) and (2.3). Let the width l of Ω be sufficiently small. If on p Q , u 0 , then we have u 0 in Q. If Ω is unbounded, and lim inf | x | u ( x , t ) 0 , then the conclusion is also true.

Proof Let 0 < ε < l ,
φ ( x , t ) = t + sin x 1 + ε l .
Then φ is positive and
φ t = 1 , a i j D i j φ = ( 1 l ) 2 a 11 φ .
Choose λ ( x , t ) = λ 0 / l 2 . In virtue of the boundedness of c ( x , t ) , when l is sufficiently small, we have c ( x , t ) > λ ( x , t ) , and thus
b φ t + a i j D i j φ λ φ = b ( 1 l ) 2 a 11 φ ( λ 0 l 2 ) φ = b ( 1 l ) 2 a 11 φ + λ 0 l 2 φ b < 0 .

From Theorem 2.1, we have u 0 . □

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 u ( x , t ) : Q R is called parabolically convex if it is continuous, convex in x and decreasing in t.

Suppose that the following conditions hold.
  1. (A)

    f u ( t , u ) / f ( t , u ) is bounded in [ 0 , T ] × R .

     
  2. (B)
    u 0 / x 1 < 0 and
    u 0 ( x ) u 0 ( x λ ) , x Ω λ ,
    (3.1)
     

where x λ = ( 2 λ x 1 , x 2 , , x n ) , Ω λ = Ω { x Ω : x 1 λ } ( λ < 0 ).

Theorem 3.1 Let Ω be a strictly convex domain in R n and symmetric with respect to the plane { x Ω : x 1 = 0 } , Q = Ω × ( 0 , T ] . Assume that conditions (A) and (B) hold and u C 2 , 1 ( Q ) C 0 ( Q ¯ ) is any parabolically convex solution of (1.1), (1.2) and (1.3). Then u ( x 1 , x , t ) = u ( x 1 , x , t ) , where ( x , t ) = ( x 1 , x , t ) R n + 1 , and when x 1 0 , u ( x , t ) / x 1 0 .

Proof Let in Ω λ × ( 0 , T ] , u λ ( x , t ) = u ( x λ , t ) , that is,
u λ ( x 1 , x 2 , , x n , t ) = u ( 2 λ x 1 , x 2 , , x n , t ) , ( x , t ) Ω λ × ( 0 , T ] .
Then
D 2 u λ ( x 1 , x 2 , , x n , t ) = P T D 2 u ( 2 λ x 1 , x 2 , , x n , t ) P ,
where P = diag ( 1 , 1 , , 1 ) . Therefore,
u t λ det ( D 2 u λ ) = u t ( 2 λ x 1 , x 2 , , x n , t ) det ( D 2 u ( 2 λ x 1 , x 2 , , x n , t ) ) = f ( t , u ( 2 λ x 1 , x 2 , , x n , t ) ) = f ( t , u λ ) .
(3.2)
We rewrite (3.2) in the form
log ( u t λ ) + log ( det ( D 2 u λ ) ) = log f ( t , u λ ) .
(3.3)
On the other hand, from (1.1), we have
log ( u t ) + log ( det ( D 2 u ) ) = log f ( t , u ) .
(3.4)
According to (3.3) and (3.4), we have
log ( u t ) log ( u t λ ) + log ( det ( D 2 u ) ) log ( det ( D 2 u λ ) ) = log f ( t , u ) log f ( t , u λ ) .
Therefore
0 1 d d s log ( s u t ( 1 s ) u t λ ) d s + 0 1 d d s log det ( s D 2 u + ( 1 s ) D 2 u λ ) d s = 0 1 d d s log f ( t , s u + ( 1 s ) u λ ) d s .
As a result, we have
b ( x , t ) ( u u λ ) t + a i j ( x , t ) ( u u λ ) i j c ( x , t ) ( u u λ ) = 0 , ( x , t ) Ω λ × ( 0 , T ] ,
(3.5)
where
b ( x , t ) = 0 1 d s s u t + ( 1 s ) u t λ , a i j ( x , t ) = 0 1 g s i j d s , c ( x , t ) = 0 1 f u f ( t , s u + ( 1 s ) u λ ) d s ,
g s i j is the inverse matrix of s D 2 u + ( 1 s ) D 2 u λ . Then b ( x , t ) < 0 , c ( x , t ) is bounded and by the a priori estimate [9] we know there exist positive constants λ 0 and Λ 0 such that
λ 0 | ξ | 2 a i j ξ i ξ j Λ 0 | ξ | 2 , ξ R n .
Let
w λ = u u λ ,
then from (3.5),
b ( x , t ) w t λ + a i j ( x , t ) w i j λ c ( x , t ) w λ = 0 , ( x , t ) Ω λ × ( 0 , T ] .
(3.6)
Clearly,
w λ ( x , t ) = 0 , x Ω λ { x 1 = λ } , 0 < t T .
(3.7)
Because the image of Ω Ω λ about the plane { x 1 = λ } lies in Ω, according to the maximum principle of parabolic Monge-Ampère equations,
u λ ( x , t ) 0 , x Ω Ω λ .
Thus
w λ ( x , t ) = u u λ = 0 u λ 0 , x Ω Ω λ , 0 < t T .
(3.8)
On the other hand, from (3.1),
w λ ( x , 0 ) = u 0 ( x ) u 0 ( x λ ) 0 , x Ω λ .
(3.9)

From Corollary 2.3, when the width of Ω λ is sufficiently small, w λ ( x , t ) 0 , ( x , t ) Ω λ × ( 0 , T ] .

Now we start to move the plane to its right limit. Define
Λ = sup { λ < 0 | w λ ( x , t ) 0 , x Ω λ , 0 < t T } .
We claim that
Λ = 0 .

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 Λ < 0 , then the image of Ω Ω Λ under the reflection about { x 1 = Λ } lies inside Ω. According to the strong maximum principle of parabolic Monge-Ampère equations, for x Ω , u Λ < 0 . Therefore, for x Ω Λ Ω , we have w Λ > 0 . On the other hand, by the definition of Λ, we have for x Ω Λ , w Λ 0 . So, from the strong maximum principle [28] of linear parabolic equations and (3.6), we have for ( x , t ) Ω Λ × ( 0 , T ] ,
w Λ ( x , t ) > 0 .
(3.10)
Let d 0 be the maximum width of narrow regions so that we can apply the narrow region principle. Choose a small positive constant δ such that Λ + δ < 0 , δ d 0 / 2 Λ . We consider the function w Λ + δ ( x , t ) on the narrow region
Σ Λ + δ × ( 0 , T ] = ( Ω Λ + δ { x 1 > Λ d 0 2 } ) × ( 0 , T ] .
Then w Λ + δ ( x , t ) satisfies
b ( x , t ) w t Λ + δ + a i j ( x , t ) D i j w Λ + δ c ( x , t ) w Λ + δ = 0 , ( x , t ) Σ Λ + δ × ( 0 , T ] .
(3.11)
Now we prove the boundary condition
w Λ + δ ( x , t ) 0 , ( x , t ) p ( Σ Λ + δ × ( 0 , T ] ) .
(3.12)
Similar to boundary conditions (3.7), (3.8) and (3.9), boundary condition (3.12) is satisfied for x Σ Λ + δ Ω , x Σ Λ + δ { x 1 = Λ + δ } and for t = 0 . In order to prove (3.12) is satisfied for x Σ Λ + δ { x 1 = Λ d 0 / 2 } , we apply the continuity argument. By (3.10) and the fact that ( Λ d 0 / 2 , x 2 , , x n ) is inside Ω Λ , there exists a positive constant c 0 such that
w Λ ( Λ d 0 2 , x 2 , , x n , t ) c 0 .
Because w λ is continuous in λ, then for small δ, we still have
w Λ + δ ( Λ d 0 2 , x 2 , , x n , t ) 0 .
Therefore boundary condition (3.12) holds for small δ. From Corollary 2.3, we have
w Λ + δ ( x , t ) 0 , x Σ Λ + δ , 0 < t T .
(3.13)
Combining (3.10) and the fact that w λ is continuous for λ, we know that w Λ + δ ( x , t ) 0 for x Ω Λ when δ is small. Then from (3.13), we know that
w Λ + δ ( x , t ) 0 , x Ω Λ + δ , 0 < t T .

This contradicts with the definition of Λ, and so Λ = 0 .

As a result, w 0 ( x , t ) 0 for x Ω 0 , which means that as x 1 < 0 ,
u ( x 1 , x 2 , , x n , t ) u ( x 1 , x 2 , , x n , t ) .
Since Ω is symmetric about the plane { x 1 = 0 } , then for x 1 0 , u ( x 1 , x 2 , , x n , t ) also satisfies (1.1). Thus we can move the plane from the right towards the left and get the reverse inequality. Therefore
u ( x , t ) / x 1 0 , x 1 0 , u ( x 1 , x 2 , , x n , t ) = u ( x 1 , x 2 , , x n , t ) .
(3.14)

Equation (3.14) means that u is symmetric about the plane { x 1 = 0 } . Theorem 3.1 is proved. □

If we put the x 1 axis in any direction, from Theorem 3.1, we have the following.

Corollary 3.2 If Ω is a ball, Q = Ω × ( 0 , T ] , then any parabolically convex solution u C 2 , 1 ( Q ¯ ) of (1.1), (1.2) and (1.3) is radially symmetric about the origin.

Remark 3.1 Solutions of (1.1) in R n × ( 0 , T ] may not be radially symmetric. For example,
u t det ( D 2 u ) = e u , ( x , t ) R n × ( 0 , T ]
(3.15)
has a non-radially symmetric solution. In fact, we know that f ( x ) = 2 log ( 1 + e 2 x ) 2 x log 4 ( x > 0 ) satisfies f = e f in R 1 , and f ( x ) = f ( x ) , x < 0 . Define
u ( x , t ) = log ( T t ) + f ( x 1 ) + f ( x 2 ) + + f ( x n ) ,

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 B = B 1 ( 0 ) be the unit ball in R n , and let radially symmetric function u 0 ( x ) = u 0 ( r ) , r = | x | satisfy
u 0 ( r ) ( u 0 ( r ) ) n 1 u 0 ( r ) r n 1 = 1 , 0 < r < 1 ,
(3.16)
u 0 ( 1 ) = u 0 ( 0 ) = 0 .
(3.17)
Example 3.1 Let u 0 satisfy (3.16) and (3.17). Then any solution of
u t det ( D 2 u ) = 1 , ( x , t ) B × ( 0 , T ] ,
(3.18)
u = 0 , ( x , t ) B × ( 0 , T ) ,
(3.19)
u = u 0 , ( x , t ) B ¯ × { 0 }
(3.20)
is of the form
u = [ ( n + 1 ) t + 1 ] 1 n + 1 u 0 ( r ) ,
(3.21)

where r = | x | .

Proof According to Corollary 3.2, the solution is symmetric. Let
u ( x , t ) = u ( r , t ) , r = | x | .
Then
u i = u ( r , t ) r x i r , u i j = 2 u ( r , t ) r 2 x i x j r 2 + u ( r , t ) r ( δ i j r x i x j r 3 ) , det ( D 2 u ) = ( u / r r ) n 1 2 u r 2 .
Therefore (3.18) is
u t ( u / r r ) n 1 2 u r 2 = 1 .
(3.22)
We seek the solution of the form
u ( r , t ) = T ( t ) u 0 ( r ) .
Then
u 0 ( r ) T ( t ) ( u 0 ( r ) T ( t ) ) n 1 r n 1 u 0 ( r ) T ( t ) = 1 .
That is,
u 0 ( r ) ( u 0 ( r ) ) n 1 u 0 ( r ) r n 1 = 1 T ( t ) ( T ( t ) ) n .
(3.23)
Therefore
T ( t ) ( T ( t ) ) n = 1 .
(3.24)
By (3.20), we know that
T ( 0 ) = 1 .
(3.25)
From (3.24) and (3.25), we have
T ( t ) = [ ( n + 1 ) t + 1 ] 1 n + 1 .
As a result,
u ( r , t ) = [ ( n + 1 ) t + 1 ] 1 n + 1 u 0 ( r ) .

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). □

Declarations

Acknowledgements

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).

Authors’ Affiliations

(1)
School of Mathematics and Information Science, Weifang University, Weifang, China

References

  1. Wang XJ: On the design of a reflector antenna. Inverse Probl. 1996, 12: 351-375. 10.1088/0266-5611/12/3/013View ArticleMATHGoogle Scholar
  2. Wang XJ: On the design of a reflector antenna II. Calc. Var. Partial Differ. Equ. 2004, 20: 329-341. 10.1007/s00526-003-0239-4View ArticleMATHGoogle Scholar
  3. Evans LC, Gangbo W Mem. Amer. Math. Soc. 137. Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem 1999.Google Scholar
  4. Gangbo W, McCann RJ: The geometry of optimal transportation. Acta Math. 1996, 177: 113-161. 10.1007/BF02392620MathSciNetView ArticleMATHGoogle Scholar
  5. Caffarelli L, Nirenberg L, Spruck J: The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Commun. Pure Appl. Math. 1984, 37: 369-402. 10.1002/cpa.3160370306MathSciNetView ArticleMATHGoogle Scholar
  6. Gutiérrez CE: The Monge-Ampère Equation. Birkhäuser, Basel; 2001.View ArticleMATHGoogle Scholar
  7. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer, Berlin; 1983.View ArticleMATHGoogle Scholar
  8. Lieberman GM: Second Order Parabolic Differential Equations. World Scientific, River Edge; 1996.View ArticleMATHGoogle Scholar
  9. Wang GL: The first boundary value problem for parabolic Monge-Ampère equation. Northeast. Math. J. 1987, 3: 463-478.MathSciNetMATHGoogle Scholar
  10. Wang RH, Wang GL: On the existence, uniqueness and regularity of viscosity solution for the first initial boundary value problem to parabolic Monge-Ampère equations. Northeast. Math. J. 1992, 8: 417-446.MathSciNetMATHGoogle Scholar
  11. Xiong JG, Bao JG: On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations. J. Differ. Equ. 2011, 250: 367-385. 10.1016/j.jde.2010.08.024MathSciNetView ArticleMATHGoogle Scholar
  12. Krylov NV: Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation. Sib. Mat. Zh. 1976, 17: 290-303. (in Russian)View ArticleMATHGoogle Scholar
  13. Firey WJ: Shapes of worn stones. Mathematika 1974, 21: 1-11. 10.1112/S0025579300005714MathSciNetView ArticleMATHGoogle Scholar
  14. Tso K: Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math. 1985, 38: 867-882. 10.1002/cpa.3160380615MathSciNetView ArticleMATHGoogle Scholar
  15. Tso K: On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations. Commun. Partial Differ. Equ. 1985, 10: 543-553. 10.1080/03605308508820388MathSciNetView ArticleMATHGoogle Scholar
  16. Gidas B, Ni WM, Nirenberg L: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 1979, 68: 209-243. 10.1007/BF01221125MathSciNetView ArticleMATHGoogle Scholar
  17. Alexandrov AD: A characteristic property of spheres. Ann. Mat. Pura Appl. 1962, 58: 303-315. 10.1007/BF02413056MathSciNetView ArticleGoogle Scholar
  18. Serrin J: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 1971, 43: 304-318.MathSciNetView ArticleMATHGoogle Scholar
  19. Li C: Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains. Commun. Partial Differ. Equ. 1991, 16: 585-615. 10.1080/03605309108820770View ArticleMATHGoogle Scholar
  20. Berestycki H, Nirenberg L: On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat. 1991, 22: 1-37. 10.1007/BF01244896MathSciNetView ArticleMATHGoogle Scholar
  21. Zhang ZT, Wang KL: Existence and non-existence of solutions for a class of Monge-Ampère equations. J. Differ. Equ. 2009, 246: 2849-2875. 10.1016/j.jde.2009.01.004View ArticleMATHGoogle Scholar
  22. Serrin J, Zou H: Symmetry of ground states of quasilinear elliptic equations. Arch. Ration. Mech. Anal. 1999, 148: 265-290. 10.1007/s002050050162MathSciNetView ArticleMATHGoogle Scholar
  23. Busca J, Sirakov B: Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 2000, 163: 41-56. 10.1006/jdeq.1999.3701MathSciNetView ArticleMATHGoogle Scholar
  24. Földes J: On symmetry properties of parabolic equations in bounded domains. J. Differ. Equ. 2011, 250: 4236-4261. 10.1016/j.jde.2011.03.018View ArticleMATHGoogle Scholar
  25. Hess P, Poláčik P: Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems. Proc. R. Soc. Edinb., Sect. A 1994, 124: 573-587. 10.1017/S030821050002878XView ArticleMATHGoogle Scholar
  26. Babin AV: Symmetrization properties of parabolic equations in symmetric domains. J. Differ. Equ. 1995, 123: 122-152. 10.1006/jdeq.1995.1159MathSciNetView ArticleMATHGoogle Scholar
  27. Babin AV, Sell GR: Attractors of non-autonomous parabolic equations and their symmetry properties. J. Differ. Equ. 2000, 160: 1-50. 10.1006/jdeq.1999.3654MathSciNetView ArticleMATHGoogle Scholar
  28. Protter MH, Weinberger HF: Maximum Principles in Differential Equations. Springer, New York; 1984.View ArticleMATHGoogle Scholar

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© Dai; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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