Steiner对称在椭圆和抛物型方程中的应用

发布时间：2018-05-03 14:06

  本文选题：椭圆型方程 + 抛物型方程　； 参考：《大连理工大学》2016年博士论文

【摘要】：重排方法已经成为研究椭圆和抛物方程的一种非常有用的工具.重排方法也称为对称化方法.Talenti首先利用重排方法研究了二阶椭圆方程.至今他的结果已经被广泛的应用和扩展.其中大部分的结果都是基于Schwarz对称来研究问题.本文主要利用Steiner对称来对偏微分方程进行研究.Schwarz对称是关于一个点的球对称递减重排,Steiner对称是关于一个超平面的对称重排,所以使用Schwarz对称来研究偏微分方程可能会丢失一些局部对称的性质.本文将利用Steiner对称来给出原问题的对称化问题,并建立原问题与对称化问题的解的比较关系.本文共分五部分：第1部分 概述重排方法在偏微分方程中的研究背景和国内外研究进展,并且简要列出本文的主要工作及相关的预备知识.第2部分 利用Steiner对称对带有零阶项的并且零阶项系数含有x的Neumann边界的椭圆方程进行了研究,利用反证法构造极值原理,得到了原问题的对称化问题是一个Dirichlet-Neumann双边界对称问题,最后建立了原问题和对称化问题的解的比较关系.第3部分 利用Steiner对称对次线性椭圆方程进行了研究,首先证明了原问题的解的L~∞估计,然后得到了次线性椭圆问题的对称化问题是一个线性问题,建立了原问题和对称化问题的解的比较关系,最后给出了两个问题的解的能量估计不等式.第4部分 利用Steiner对称对次线性抛物方程问题进行了研究,首先证明了原问题的解的L~∞估计,然后得到了抛物问题的对称化问题,建立了原问题和对称化问题之间的解的比较关系,给出了两个问题的解的能量估计不等式.第5部分 给出结论与展望.
[Abstract]:The rearrangement method has become a very useful tool for the study of elliptic and parabolic equations. The rearrangement method is also called symmetry method. Talenti first studies the second order elliptic equation by using the rearrangement method. So far his results have been widely used and expanded. Most of the results are based on Schwarz symmetry. In this paper, we mainly use Steiner symmetry to study partial differential equations. Schwarz symmetry is about the spherical symmetry decline rearrangement of a point and Steiner symmetry is about a hyperplane symmetric rearrangement. So using Schwarz symmetry to study partial differential equations may lose some properties of local symmetry. In this paper, the symmetry problem of the original problem is given by using Steiner symmetry, and the comparison between the solution of the original problem and the symmetric problem is established. This paper is divided into five parts: the first part summarizes the research background of rearrangement method in partial differential equations and the research progress at home and abroad, and briefly lists the main work and related preparatory knowledge of this paper. In the second part, the Steiner symmetry is used to study the elliptic equations with zero order term and the coefficient of zero order term with Neumann boundary x, and the extreme value principle is constructed by using the counter proof method. It is obtained that the symmetry problem of the original problem is a Dirichlet-Neumann double boundary symmetric problem. Finally, a comparative relationship between the solution of the original problem and the symmetric problem is established. In the third part, the sublinear elliptic equation is studied by using Steiner symmetry. Firstly, the L 鈭，

本文编号：1838789