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Eigenvalues and eigenfunctions of double layer potentials


Authors: Yoshihisa Miyanishi and Takashi Suzuki
Journal: Trans. Amer. Math. Soc. 369 (2017), 8037-8059
MSC (2010): Primary 47G40; Secondary 34L20
DOI: https://doi.org/10.1090/tran/6913
Published electronically: May 1, 2017
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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

$\displaystyle (K \psi )(x) \equiv \int _{\partial \Omega } \psi (y)\cdot \nu _{y} E(x, y) \; ds_y, $

where

$\displaystyle E(x, y)= \begin {cases}\frac {1}{\pi } \log \frac {1}{\vert x-y\v... ...\frac {1}{\vert x-y\vert}, \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 $.

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Additional Information

Yoshihisa Miyanishi
Affiliation: Center for Mathematical Modeling and Data Science, Osaka University, Toyonaka 560-8531, Japan
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
Email: suzuki@sigmath.es.osaka-u.ac.jp

DOI: https://doi.org/10.1090/tran/6913
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.
Article copyright: © Copyright 2017 American Mathematical Society