https://scholars.lib.ntu.edu.tw/handle/123456789/30367
Title: | 節點域定理和相關的主題 Nodal Domain Theorem and Related Topics |
Authors: | 謝昇諺 Hsieh, Sheng-Yen |
Keywords: | 斯特克羅夫;特徵值問題;節點域定理;Stekloff;Steklov;eigenvalue problem;nodal domain | Issue Date: | 2009 | Abstract: | 這篇文章介紹了節點域定理。對於調和函數的特徵值問題,第N個特徵函數的節點域個數K(u_N),小於或等於N. 對於二階橢圓特徵值問題,當維度d大於等於3且主要係數A是Holder連續時,K(u_N) 小於等於 2(N-1)。對於二階橢圓Stekloff特徵值問題,當d = 2且A是L^1或是d大於等於3且A是Lipschitz時,K(u_N)小於等於N。對於雙調和函數的特徵值問題,當d = 1,K(u_N)小於等於N. 然而,對於d大於等於2,這一般不會成立。最後,我們用Krein-Rutman定理來討論主要特徵函數的同號性。 This article introduces the nodal domain theorem. For harmonic eigenvalue problem, the number of nodal domain of N-th eigenfunction, K(u_N), less than N. For second orderlliptic eigenvalue problem, when dimension d is greater than or equal to 3 and the principal coeffcient A is Holderontinuous, K(u_N) is less than or equal to 2(N-1). For second order elliptic Stekloff eigenvalue problem, when = 2 and A is L^1 or d is greater than or equal to 3 and A 2 is Lipschitz, K(u_N) is less than or equal to N. For biharmonic eigenvalue problem, when d = 1, K(u_N) is less than or equal to N. However, it generally not holds for d is greater than or equal to 2. Finally, we use Krein-Rutman theorem to discuss the one-sign property of principal eigenfunction. |
URI: | http://ntur.lib.ntu.edu.tw//handle/246246/180658 |
Appears in Collections: | 數學系 |
File | Description | Size | Format | |
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ntu-98-R96221013-1.pdf | 23.53 kB | Adobe PDF | View/Open |
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