Eigenvalues and eigenfunctions of double layer potentials
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- by Yoshihisa Miyanishi and Takashi Suzuki PDF
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Abstract:
Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let $\Omega$ be a $C^2$ bounded region in ${\mathbf {R}}^n$ ($n=2, 3$). The double layer potential $K: L^2(\partial \Omega ) \rightarrow L^2(\partial \Omega )$ is defined by \[ (K \psi )(x) \equiv \int _{\partial \Omega } \psi (y)\cdot \nu _{y} E(x, y) \; ds_y, \] where \[ E(x, y)= \begin {cases} \frac {1}{\pi } \log \frac {1}{|x-y|}, \quad \;\mbox {if}\; n=2, \\ \frac {1}{2\pi } \frac {1}{|x-y|}, \quad \hspace {4mm}\;\mbox {if} \; n=3, \end {cases} \] $ds_y$ is the line or surface element and $\nu _y$ is the outer normal derivative on $\partial \Omega$. It is known that $K$ is a compact operator on $L^2(\partial \Omega )$ and consists of at most a countable number of eigenvalues, with $0$ as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of $\partial \Omega$.References
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Additional Information
- Yoshihisa Miyanishi
- Affiliation: Center for Mathematical Modeling and Data Science, Osaka University, Toyonaka 560-8531, Japan
- MR Author ID: 633586
- ORCID: 0000-0002-8252-4267
- Email: miyanishi@sigmath.es.osaka-u.ac.jp
- Takashi Suzuki
- Affiliation: Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531, Japan
- MR Author ID: 199324
- Email: suzuki@sigmath.es.osaka-u.ac.jp
- Received by editor(s): January 15, 2015
- Received by editor(s) in revised form: December 22, 2015
- Published electronically: May 1, 2017
- Additional Notes: This work was supported partly by JSPS Grant-in-Aid for Scientific Research (A) 26247310.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 8037-8059
- MSC (2010): Primary 47G40; Secondary 34L20
- DOI: https://doi.org/10.1090/tran/6913
- MathSciNet review: 3695853