Journal article
A generalized eigenproblem for the Laplacian which arises in lightning
Journal of mathematical analysis and applications, v 341(2), pp 1028-1041
15 May 2008
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Abstract
The following generalized eigenproblem is analyzed: Find u is an element of H-0(1) (Omega), u not equal 0, and lambda is an element of R such that
<del u, del v >(D) = lambda <del u, del v >(Omega)
for all V is an element of H-0(1) (Omega), where Omega subset of R-n is a bounded domain, D is a subdomain with closure contained in Omega, and <., .>(Omega) is the inner product
<del u, del v >(Omega) = integral(Omega) del u.del v dx.
It is proved that any f is an element of H-0(1) (Omega) can be expanded in terms of orthogonal eigenfunctions for the generalized eigenproblem. During the analysis, we present a new inner product on H-1/2(partial derivative D) with the following properties: (a) the norm associated with the inner product is equivalent to the usual norm on H-1/2 (partial derivative D), and (b) the double layer potential operator is self adjoint with respect to the new inner product and compact as a mapping from H-1/2(partial derivative D) into itself. The analysis identifies four classes of eigenfunctions for the generalized eigenproblem:
1. The function Pi which is 1 on D and harmonic on Omega \ D; the eigenvalue is 0.
2. Functions in H-0(1) (Omega) with support in Q \ D; the eigenvalue is 0.
3. Functions in H-0(1) (Omega) with support in D; the eigenvalue is 1.
4. Excluding Pi, the harmonic extension of the eigenfunctions of a double layer potential on a D. The eigenvalues are contained in the open interval (0, 1). The only possible accumulation point is lambda = 1/2.
A positive lower bound for the smallest positive eigenvalue is obtained. These results can be used to evaluate the change in the electric potential due to a lightning discharge.
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Details
- Title
- A generalized eigenproblem for the Laplacian which arises in lightning
- Creators
- Beyza Caliskan Aslan - Univ Florida, Dept Math, Gainesville, FL 32611 USAWilliam W. Hager - Univ Florida, Dept Math, Gainesville, FL 32611 USAShari Moskow - Univ Florida, Dept Math, Gainesville, FL 32611 USA
- Publication Details
- Journal of mathematical analysis and applications, v 341(2), pp 1028-1041
- Publisher
- Elsevier
- Number of pages
- 14
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000254588300023
- Scopus ID
- 2-s2.0-38949093647
- Other Identifier
- 991021863512704721
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- Web of Science research areas
- Mathematics
- Mathematics, Applied