臺灣博碩士論文加值系統

移動平面法和滑動法是處理橢圓偏微分方程的技巧，它們可以簡單而有效地證明解的對稱性或沿某一方向的遞增性。它們都以「橢圓方程極值定理在小區域上自動成立」為基礎。
我們將照著Beresycki和Nirenberg在"On the method of moving planes and the sliding mehtod" 文章中的討論，把這些方法應用於非線性橢圓方程在可弱微分兩次的sobolev 函數空間中的解；但B&N 文中只研究古典解的情形，且對討論的區域有限制 (譬如，對某一方向是凸的) 。在討論解對某一方向的遞增性時，已可知區域不需有限制。但為了放鬆解空間的限制，我們發現某種有趣形式(橢圓和拋物線型混合方程)的極值定理是關鍵，但目前還未證明之。
最後，我們也把這些方法應用在橢圓方程組上。例如合作模型，及如何避免競爭模型等的討論，這部份還有更多可以探討之處。

The Method of Moving Planes and the Sliding Method are simple and powerful techniques in proving the symmetry and monotonicity in, say, the $x_1$ direction for a solution of an
elliptic equation. They rely on the "Maximum Principle in Small Domain."
Following a discussion similar to that in "On the method of moving planes and the sliding mehtod" by Beresycki and Nirenberg, we apply the methods to $u \in W_{loc}^{2,n+1}(\Omega ) \cap C^0(\overline{\Omega})$ which satisfies the nonlinear elliptic equation $F(x, u, Du, D^2u) = 0$ in an arbitrary bounded domain $\Omega$ in $\mathbf{R}^n$. However, Berestycki and Nirenberg assumed that $u \in C^2(\Omega ) \cap C^0(\overline{\Omega})$ and $\Omega$ is convex in the $x_1$-direction. We have successfully loosen the later assumption, but while dealing with the former, some interesting type of Maximum Principle was required and had yet to be proved.
We also applied the methods on some simple system elliptic equations which contains much more to be discussed.

Abstract
1 Preliminary Knowledge
1.1 Alexandroff Maximum Principle
1.2 Some Theorems about Regularity
1.3 The Methods for Simple Equations in General Domain
2 Methods Applied to Fully Nonlinear Equations
2.1 Main Results
2.2 Maximum Principle for Parabolic Equations
2.3 Proof of Main Results
3 Application to System Partial Differential Equations
3.1 Cooperate type
3.2 Conditionss to Rule Out the Competitive Case
3.3 The Case where u_1, u_2,...,u_k are Linear Dependent
Bibliography

H.Beresycki; L.Nirenberg.
On the method of moving planes and the sliding method.
Bol. Soc. Brasil. Mat. (N.S.){22} (1991), no.1, 1-37.
D.Gilbarg; N.S.Trudinger.
Elliptic Partial Differential Equations of Second Order,
2nd ed. Spring-Verlag, Berlin Heidelberg, 1983.
Gary M.Lieberman.
Second Order Parabolic Differential Equations.
World Scientific Publishing Co. Pte. Ltd. Singapore, 1996.