臺灣博碩士論文加值系統

在旋轉曲面上S，考慮Laplace算子固有問題：
Δu+λu=0 u=0在S的邊界.
我們利用分離變數法把Laplace算子轉換成Sturm-Liouville算子，再利用變數變換得到第一固有值與區域所在位置的函數關係。
對於旋轉曲面上，在帶狀區域面積固定為A時，固有的大小關係和區域所在位置有關。我們得到下列結果：當生成函數為f(x)=ax+b>0，若a>0，隨著等面積帶狀區域向右移動，則固有跟著增加；若a<0，隨著等面積帶狀區域向右移動，則固有跟著減少。當生成函數為f(x)=x^3>0，隨著等面積帶狀區域向右移動，則固有跟著增加。

Consider the eigenvalue of the Laplace operator on the surfaces of revolution S：
Δu+λu=0 u=0 on boundary S.
Using separation of variable, we can transform a Laplace operator into a Sturm-Liouville operator. Through the study on the Sturm-Liouville eigenvalue problem, we obtain the variation of the first eigenvalues of the Laplace operator.According to the location of the band region with fixed area on S.
Indeed, we obtain the following results：When the generating function is f(x)=ax+b&gt;0 .IF a&gt;0, the first eigenvalue is increasing as the band region moves to the right side. Otherwise, the first eigenvalue is decreasing. When the generating function is f(x)=x^3&gt;0 . The first eigenvalue is increasing as the band region moves to the right side.

目錄
中文摘要.............................................. 1
英文摘要.............................................. 2
第一章 前言.......................................... 6
Laplace固有簡介................................ 6
Sturm-Liouville算子的固有問題.................. 8
Laplace算子與Sturm-Liouville算子固有值的關係.... 9

第二章 導論......................................... 13
線性代數上的固有值問題........................... 13
微分方程上的固有值問題........................... 14
旋轉面上的固有值問題............................. 15

第三章 固有值的單調性............................... 18
生成函數為 固有之性質.................... 20
生成函數為 固有之性質......................... 21

第四章 討論.......................................... 23

參考文獻 ............................................. 24

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