移动平面法的积分形式对一类非线性偏微分方程（组）的应用

发布时间：2019-03-27 15:15

【摘要】：本文分别在有界区域、无界外部区域和Rn空间的上半部分研究一类非线性偏微分方程(组)解的单调性对称性及在无界范围上解的Liouville型定理。一般使用移动平面法研究解的单调性和对称性。传统的移动平面法适用于偏微分方程,需要用到微分方程的极值原理或标准的边界点引理这一类局部性质,然而对本论文所研究的非线性偏微分方程,并没有找到在相应区域上的极值原理,为了解决这一困难,我们使用移动平面法的积分形式来代替传统的移动平面法。首先,我们将这类非线性偏微分方程(组)转化为相应的积分方程(组)。这些积分方程(组)的解满足的性质同样也适用于相应的非线性偏微分方程(组)的解。然后应用移动平面法的积分形式研究所得到的积分方程(组)。在第一章,我们主要是在有界区域上研究一类抽象的非线性偏微分方程解的单调性和对称性以及该有界区域的对称性。在第二章,主要研究在无界外区域上一类非线性偏微分方程组解的单调性和对称性。在第三章,主要研究在无界外区域上另一个相对较具体的非线性偏微分方程组解的单调性和对称性,从而进一步得到相应解所满足的Liouville型定理。在第四章,我们主要研究在Rn空间的上半部分一类分数阶偏微分方程组的解所满足的Liouville型定理。
[Abstract]:In this paper, the monotonicity symmetry of solutions of a class of nonlinear partial differential equations and the Liouville type theorem of solutions in unbounded range are studied in the bounded domain, the unbounded outer domain and the upper half of the Rn space respectively. Moving plane method is generally used to study the monotonicity and symmetry of solutions. The traditional moving plane method is suitable for partial differential equations. It is necessary to use the extreme value principle of differential equations or the standard boundary point Lemma. However, for the nonlinear partial differential equations studied in this paper, In order to solve this problem, we use the integral form of the moving plane method instead of the traditional moving plane method. First, we convert this kind of nonlinear partial differential equations into corresponding integral equations. The properties of the solutions of these integral equations (groups) are also applicable to the solutions of the corresponding nonlinear partial differential equations (systems). Then the integral equation (group) obtained by using the integral form of the moving plane method is studied. In the first chapter, we mainly study the monotonicity and symmetry of solutions of a class of abstract nonlinear partial differential equations on the bounded domain and the symmetry of the bounded domain. In the second chapter, we mainly study the monotonicity and symmetry of solutions of a class of nonlinear partial differential equations on an unbounded outer domain. In the third chapter, we mainly study the monotonicity and symmetry of the solutions of another relatively specific system of nonlinear partial differential equations on the unbounded outer domain, and then obtain the Liouville type theorem of the corresponding solutions. In chapter 4, we mainly study the Liouville type theorem for the solutions of a class of fractional order partial differential equations in the upper half of Rn space.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.29

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