本文選題：橢圓型方程 + 拋物型方程�。� 參考：《大連理工大學》2016年博士論文

【摘要】：重排方法已經(jīng)成為研究橢圓和拋物方程的一種非常有用的工具.重排方法也稱為對稱化方法.Talenti首先利用重排方法研究了二階橢圓方程.至今他的結果已經(jīng)被廣泛的應用和擴展.其中大部分的結果都是基于Schwarz對稱來研究問題.本文主要利用Steiner對稱來對偏微分方程進行研究.Schwarz對稱是關于一個點的球?qū)ΨQ遞減重排,Steiner對稱是關于一個超平面的對稱重排,所以使用Schwarz對稱來研究偏微分方程可能會丟失一些局部對稱的性質(zhì).本文將利用Steiner對稱來給出原問題的對稱化問題,并建立原問題與對稱化問題的解的比較關系.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國內(nèi)外研究進展,并且簡要列出本文的主要工作及相關的預備知識.第2部分 利用Steiner對稱對帶有零階項的并且零階項系數(shù)含有x的Neumann邊界的橢圓方程進行了研究,利用反證法構造極值原理,得到了原問題的對稱化問題是一個Dirichlet-Neumann雙邊界對稱問題,最后建立了原問題和對稱化問題的解的比較關系.第3部分 利用Steiner對稱對次線性橢圓方程進行了研究,首先證明了原問題的解的L~∞估計,然后得到了次線性橢圓問題的對稱化問題是一個線性問題,建立了原問題和對稱化問題的解的比較關系,最后給出了兩個問題的解的能量估計不等式.第4部分 利用Steiner對稱對次線性拋物方程問題進行了研究,首先證明了原問題的解的L~∞估計,然后得到了拋物問題的對稱化問題,建立了原問題和對稱化問題之間的解的比較關系,給出了兩個問題的解的能量估計不等式.第5部分 給出結論與展望.
[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，

本文編號：1838789