Open Access Research

Symmetry of solutions to parabolic Monge-Ampère equations

Limei Dai

Author Affiliations

School of Mathematics and Information Science, Weifang University, Weifang, Shandong, 261061, China

Boundary Value Problems 2013, 2013:185  doi:10.1186/1687-2770-2013-185


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/185


Received:3 April 2013
Accepted:5 August 2013
Published:20 August 2013

© 2013 Dai; licensee Springer

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.

Abstract

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M1">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M2">View MathML</a> 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 [5-7] and the study of parabolic Monge-Ampère equations; see the references [8-11]etc. The parabolic Monge-Ampère equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M3">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M3">View MathML</a> is the most appropriate parabolic version of the elliptic Monge-Ampère equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M5">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M6">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M7">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M8">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M9">View MathML</a> is the Hessian matrix of u in x, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M10">View MathML</a>, Ω is a bounded and convex open subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M12">View MathML</a> denotes the side of Q, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M13">View MathML</a> denotes the bottom of Q, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M14">View MathML</a> denotes the parabolic boundary of Q, f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M15">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M16">View MathML</a> or Ω is a smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11">View MathML</a>, convex in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M18">View MathML</a> and symmetric with respect to the hyperplane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M19">View MathML</a>, then any positive solution of the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M20">View MathML</a>

satisfies the following symmetry and monotonicity properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M21">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M22">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M23">View MathML</a> and they got the following results.

Let Ω be a bounded convex domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11">View MathML</a> with smooth boundary and symmetric with respect to the hyperplane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M25">View MathML</a>, then each solution of the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M26">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M28">View MathML</a>, Gidas et al.[16] studied parabolic equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M30">View MathML</a>, and they proved that parabolic equations possessed the same symmetry as the above elliptic equations. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M31">View MathML</a>, Hess and Poláčik [25] first studied the asymptotic symmetry results for classical, bounded, positive solutions of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M32">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M33">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M34">View MathML</a> is as follows.

Let Ω be convex and symmetric in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M18">View MathML</a>. If u is a bounded positive solution of (1.6) and (1.7) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M34">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M37">View MathML</a>

then u has the symmetry and monotonicity properties for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M38">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M39">View MathML</a>

The result of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M31">View MathML</a> is as follows.

Assume that u is a bounded positive solution of (1.6) and (1.7) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M31">View MathML</a> such that for some sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M42">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M43">View MathML</a>

Then u is asymptotically symmetric in the sense that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M44">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48">View MathML</a> be continuous functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M49">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48">View MathML</a> is bounded and there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M53">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M54">View MathML</a>

Here and in the sequel, we always denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M55">View MathML</a>

We use the standard notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M56">View MathML</a> to denote the class of functions u such that the derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M57">View MathML</a> are continuous in Q for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M58">View MathML</a>.

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M59">View MathML</a>be a bounded continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M60">View MathML</a>, and let the positive function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M61">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M62">View MathML</a>

(2.1)

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M63">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M64">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M65">View MathML</a>

(2.3)

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M66">View MathML</a>

(2.4)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67">View MathML</a>inQ.

Proof We argue by contradiction. Suppose there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M68">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M69">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M70">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M71">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M72">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M73">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M74">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M75">View MathML</a> attains its minimum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M76">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M78">View MathML</a>. In addition, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M79">View MathML</a>. A direct calculation gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M80">View MathML</a>

Taking into account <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M81">View MathML</a>, we have at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M82">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M83">View MathML</a>

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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M84">View MathML</a>. Besides the conditions of Theorem 2.1, we assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M85">View MathML</a>

(2.5)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67">View MathML</a>inQ.

Proof Still consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M87">View MathML</a> in the proof of Theorem 2.1. Condition (2.5) shows that the minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M87">View MathML</a> 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,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M89">View MathML</a>

then we have the following narrow region principle.

Corollary 2.3 (Narrow region principle)

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M63">View MathML</a>satisfies (2.2) and (2.3). Let the widthlof Ω be sufficiently small. If on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67">View MathML</a>, then we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67">View MathML</a>in Q. If Ω is unbounded, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M94">View MathML</a>, then the conclusion is also true.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M95">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M96">View MathML</a>

Then φ is positive and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M97">View MathML</a>

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M98">View MathML</a>. In virtue of the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48">View MathML</a>, when l is sufficiently small, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M100">View MathML</a>, and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M101">View MathML</a>

From Theorem 2.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M67">View MathML</a>. □

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M103">View MathML</a> is called parabolically convex if it is continuous, convex in x and decreasing in t.

Suppose that the following conditions hold.

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M104">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M105">View MathML</a>.

(B) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M106">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M107">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M109">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M110">View MathML</a>).

Theorem 3.1Let Ω be a strictly convex domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11">View MathML</a>and symmetric with respect to the plane<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M84">View MathML</a>. Assume that conditions (A) and (B) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M63">View MathML</a>is any parabolically convex solution of (1.1), (1.2) and (1.3). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M115">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M116">View MathML</a>, and when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M118">View MathML</a>.

Proof Let in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M120">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M121">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M122">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M123">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M124">View MathML</a>

(3.2)

We rewrite (3.2) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M125">View MathML</a>

(3.3)

On the other hand, from (1.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M126">View MathML</a>

(3.4)

According to (3.3) and (3.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M127">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M128">View MathML</a>

As a result, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M129">View MathML</a>

(3.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M130">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M131">View MathML</a> is the inverse matrix of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M132">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M48">View MathML</a> is bounded and by the a priori estimate [9] we know there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M53">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M137">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M138">View MathML</a>

then from (3.5),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M139">View MathML</a>

(3.6)

Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M140">View MathML</a>

(3.7)

Because the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M141">View MathML</a> about the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M142">View MathML</a> lies in Ω, according to the maximum principle of parabolic Monge-Ampère equations,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M143">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M144">View MathML</a>

(3.8)

On the other hand, from (3.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M145">View MathML</a>

(3.9)

From Corollary 2.3, when the width of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M146">View MathML</a> is sufficiently small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M148">View MathML</a>.

Now we start to move the plane to its right limit. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M149">View MathML</a>

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M150">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M151">View MathML</a>, then the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M152">View MathML</a> under the reflection about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M153">View MathML</a> lies inside Ω. According to the strong maximum principle of parabolic Monge-Ampère equations, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M155">View MathML</a>. Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M156">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M157">View MathML</a>. On the other hand, by the definition of Λ, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M159">View MathML</a>. So, from the strong maximum principle [28] of linear parabolic equations and (3.6), we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M160">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M161">View MathML</a>

(3.10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M162">View MathML</a> be the maximum width of narrow regions so that we can apply the narrow region principle. Choose a small positive constant δ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M163">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M164">View MathML</a>. We consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M165">View MathML</a> on the narrow region

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M166">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M165">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M168">View MathML</a>

(3.11)

Now we prove the boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M169">View MathML</a>

(3.12)

Similar to boundary conditions (3.7), (3.8) and (3.9), boundary condition (3.12) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M171">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M172">View MathML</a>. In order to prove (3.12) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M173">View MathML</a>, we apply the continuity argument. By (3.10) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M174">View MathML</a> is inside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M175">View MathML</a>, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M176">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M177">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M178">View MathML</a> is continuous in λ, then for small δ, we still have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M179">View MathML</a>

Therefore boundary condition (3.12) holds for small δ. From Corollary 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M180">View MathML</a>

(3.13)

Combining (3.10) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M178">View MathML</a> is continuous for λ, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M182">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M158">View MathML</a> when δ is small. Then from (3.13), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M184">View MathML</a>

This contradicts with the definition of Λ, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M185">View MathML</a>.

As a result, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M186">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M187">View MathML</a>, which means that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M188">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M189">View MathML</a>

Since Ω is symmetric about the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M190">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M192">View MathML</a> also satisfies (1.1). Thus we can move the plane from the right towards the left and get the reverse inequality. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M193">View MathML</a>

(3.14)

Equation (3.14) means that u is symmetric about the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M190">View MathML</a>. Theorem 3.1 is proved. □

If we put the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M18">View MathML</a> axis in any direction, from Theorem 3.1, we have the following.

Corollary 3.2If Ω is a ball, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M10">View MathML</a>, then any parabolically convex solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M197">View MathML</a>of (1.1), (1.2) and (1.3) is radially symmetric about the origin.

Remark 3.1 Solutions of (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M2">View MathML</a> may not be radially symmetric. For example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M199">View MathML</a>

(3.15)

has a non-radially symmetric solution. In fact, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M200">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M201">View MathML</a>) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M202">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M203">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M205">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M206">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M207">View MathML</a> be the unit ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M11">View MathML</a>, and let radially symmetric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M210">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M211">View MathML</a>

(3.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M212">View MathML</a>

(3.17)

Example 3.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M15">View MathML</a> satisfy (3.16) and (3.17). Then any solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M214">View MathML</a>

(3.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M215">View MathML</a>

(3.19)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M216">View MathML</a>

(3.20)

is of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M217">View MathML</a>

(3.21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M210">View MathML</a>.

Proof According to Corollary 3.2, the solution is symmetric. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M219">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M220">View MathML</a>

Therefore (3.18) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M221">View MathML</a>

(3.22)

We seek the solution of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M222">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M223">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M224">View MathML</a>

(3.23)

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M225">View MathML</a>

(3.24)

By (3.20), we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M226">View MathML</a>

(3.25)

From (3.24) and (3.25), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M227">View MathML</a>

As a result,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M228">View MathML</a>

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 Middle-Aged Scientists Research Awards Fund (BS2011SF025), Shandong Province Science and Technology Development Project (2011YD16002).

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