臺灣博碩士論文加值系統

本論文關心的是半線性橢圓方程組解的對稱性。本論文在Berestycki, H.和Nirenberg, L.的研究基礎上，將解的對稱性推廣至半線性橢圓方程組。如果定義域是有限的，且在某方向上是凸的(convex)，那麼，經由移動平面法(the method of moving planes)，我們可以得出解在那個方向上的對稱性。不僅如此，假若定義域是球，並且已知解(u,v)是球對稱的(radially symmetric)，那麼我們可以找出u和v的關係。如果把這個關係代入原本的橢圓方程組，那麼原本的方程組就再也不是方程組了，而會被化約成單一橢圓方程式。因此，若是此化約的單一方程式無解，我們就可以確定原本的半線性橢圓方程組沒有球對稱解。

In this paper, we are interested in the symmetry of solutions of systems os semilinear elliptic equations. We follow the results discovered by Berestycki, H. and Nirenberg, L. and generalize to systems of semilinear elliptic equations. By using the method of moving planes, we can get the symmetry in some direction for solutions of systems of semilinear elliptic equations in a bounded domain which is convex in that direction. Furthermore, if the domain is a ball and the solution (u,v) is radially symmetric, then we can find some relation between u and v. If we apply this relation to the original system of semilinear elliptic equations, it can be reduced to a single elliptic equation. So if this reduced single elliptic equation has no solutions, we are sure that the original system of semilinear elliptic equations has no radially symmetric solutions.

第一章 緒論
第二章 極大值原理
第三章 單一非線性橢圓方程式的一些研究結果
第四章 半線性橢圓方程組
Chapter 1 Introduction
Chapter 2 Maximum Principles
Chapter 3 Some Results in Single Nonlinear Elliptic Equations
Chapter 4 Systems of Semilinear Elliptic Equations

[1] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd Edition. Springer Verlag, 1983
[2] Berestycki, H. ; L. Nirenberg, On the method of moving planes and the sliding method. Bol.Soc.Brasil.Mat(N.S.) 22 (1991), no.1,1-37 (Reviewer : C.A.Swanson)
[3] B.Gidas, W.M.Ni, L. Nirenberg, Symmetry and related properties via the maximum principles, Comm.Math.Phys.68 (1979)209-243
[4] Quing Han; Fanghua Lin, Elliptic Partial Differential Equations (Lecture Notes). New York University.
[5] Ya-Ju Jocephy Tsai, Master thesis The method of moving planes and sliding method applied to elliptic partial differential equations