Nodal Domain Theorem and Related Topics
Date Issued
2009
Date
2009
Author(s)
Hsieh, Sheng-Yen
Abstract
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.
Subjects
Stekloff
Steklov
eigenvalue problem
nodal domain
Type
thesis
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