節點域定理和相關的主題

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 order elliptic eigenvalue problem, when dimension d is greater than or equal to 3 and the principal coeffcient A is Holder continuous, K(u_N) is less than or equal to 2(N-1). For second order elliptic Stekloff eigenvalue problem, when d = 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.